Extremal length functions are log-plurisubharmonic
Hideki Miyachi
In this paper, we show that the extremal length functions on Teichm\"uller space are log-plurisubharmonic. As a corollary, we obtain an alternative proof of L.Liu and W.Su's results on the plurisubharmonicity of extremal length functions. We also obtain alternative proofs of S.Krushkal's results that a function defined by the Teichm\"uller distance from a reference point is plurisubharmonic, and the Teichm\"uller space is hyperconvex. To show the log-plurisubharmonicity, we give an explicit formula of the Levi form of the extremal length functions in generic case.
Extremal_length_function.tex
\documentclass[dvipdf]{amsart}

\title[]{Extremal length functions are log-plurisubharmonic}
\author{Hideki Miyachi}
\address{Department of Mathematics,
Graduate School of Science,
Osaka University,
Machikaneyama 1-1, Toyonaka, Osaka 560-0043, Japan}
\subjclass[2010]{32G15, 30F45, 31C10}
\keywords{Teichm\"uller theory, Teichm\"uller distance, Extremal length, Plurisubharmonic functions}

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\begin{document}
\maketitle

\begin{abstract}
In this paper,
we show that the extremal length functions on Teichm\"uller space
are log-plurisubharmonic.
As a corollary,
we obtain an alternative proof of L.Liu and W.Su's results
on the plurisubharmonicity of extremal length functions.
We also obtain alternative proofs of S.Krushkal's results that
a function defined by the Teichm\"uller distance
from a reference point
is plurisubharmonic,
and the Teichm\"uller space is hyperconvex.
To show the log-plurisubharmonicity,
we give an explicit formula of the Levi form of the extremal length functions
in generic case.
\end{abstract}

\tableofcontents


\section{Introduction}
\subsection{Results}
A positive function $U$ on a complex manifold $N$ is said to be
\emph{log-plurisubharmonic} if
$\log U$ is plurisubharmonic.
Any log-plurisubharmonic function is plurisubharmonic
(cf. \S\ref{subsec:log-plurisubharmonicity}).

Let $\teich{g,m}$ be the Teichm\"uller space of Riemann surfaces
of analytically finite type $(g,m)$ with $2g-2+m>0$.
One of the aims of this paper is to give
the following theorem,
which improves
a result by L.Liu and W. Su
in \cite{2012arXiv1210.0743L}.

\begin{theorem}[Log-plurisubharminicity]
\label{thm:pluri-subharmonicity}
Extremal length functions on $\teich{g,m}$
are log-plurisubharmonic.
\end{theorem}
Moreover,
we will also observe that
for generic measured foliations,
the log-extremal length functions are real analytic strictly plurisubharmonic functions on $\teich{g,m}$
(cf. Theorem \ref{thm:strong-positivity2-2}).

When $3g-3+m=1$,
Theorem \ref{thm:pluri-subharmonicity} follows
from the direct calculation
(cf. \S\ref{subsec:examples}).
Hence,
in the discussion in the later section,
we always assume that $3g-3+m\ge 2$.
We will prove Theorem \ref{thm:pluri-subharmonicity}
in \S\ref{subsec:log-plurisubharmonicity}.
In fact,
Theorem \ref{thm:pluri-subharmonicity} is derived from
a more stronger property of the extremal length functions
than that given in Theorem \ref{thm:pluri-subharmonicity}
(cf. Theorem \ref{thm:plurisuperharmonicitiy-of-reciprocals}).
However,
the property in Theorem \ref{thm:pluri-subharmonicity}
is also important as we see below
(cf. Corollaries \ref{coro:plurisubharmoncity-of-positive-polynomial}
and \ref{coro:Teichmuller distance}).
%
%Indeed,
%we show that
%when a measured foliation $F$ 
%is essentially complete,
%%has only three prong singularities and one prong singularities at punctures,
%%and does not contain saddle connections,
%the log-extremal length function
%$$
%\teich{g,m}\ni x\mapsto \log\ext_{x}(F)
%$$
%is a real analytic striclty plurisubharmonic function
%(cf. \S\ref{subsec:measured-foliatons}).
%Furthermore,
To show Theorem \ref{thm:pluri-subharmonicity},
we will give an explicit formula of the Levi forms of
extremal length functions
in generic case.
(cf. Theorem \ref{thm:leviform}).

The following immediately follows from Theorem \ref{thm:pluri-subharmonicity} (cf. \cite[Colollary 2.6.9]{MR1150978}).

\begin{corollary}[Log-plurisubharmonicity of positive polynomials]
\label{coro:plurisubharmoncity-of-positive-polynomial}
Let $F_{1},\cdots,F_{n}$ be measured foliations.
For any polynomial $P$ of $n$-variables with positive coefficients,
the function
$$
\teich{g,m}\ni x\mapsto
P(
\ext_{x}(F_{1}),\cdots,\ext_{x}(F_{n})
)
\in \mathbb{R}
$$
is log-plurisubharmonic.
\end{corollary}

\subsection{Teichm\"uller distance is plurisubharmonic}
%The Teichm\"uller space $\teich{g,m}$ is biholomorphically equivalent
%to a bounded domain in $\mathbb{C}^{3g-3+m}$
%(cf. \cite[Chapter 6]{MR1215481}).
%Take two essentially complete measured foliations.
%%$F$ and $G$ which have only three prong singularities
%%and one prong singularities at punctures
%%and contain no saddle connections.
%Suppose that they are transverse in the sense that
%\begin{equation}
%\label{eq:transverse-mf}
%i(F,H)+i(G,H)>0
%\end{equation}
%for all $H\in \mathcal{MF}-\{0\}$.
%From Minsky's inequality,
%the sum
%\begin{equation}
%\label{eq:extremal-length-exhausion}
%\teich{g,m}\ni x\mapsto \ext_{x}(F)+\ext_{x}(G)
%\end{equation}
%is a positive, proper, real analytic and strictly plurisubharmonic function
%(cf. \S\ref{Kerckhoff-Minsky}).
%By virtue of Oka's theorem,
%we have the following result,
%which is first proven by L.Bers and L.Ehrenpreis
%(see \cite{MR0168800}.
%See also \citeMR0352450,
%\cite{2012arXiv1210.0743L},
%\cite{MR766636}, 
%\cite{MR880186}).
%
%\begin{corollary}[Holomorphic convexity]
%\label{coro:stein}
%The Teichm\"uller space $\teich{g,m}$ is
%a domain of holomorphy.
%\end{corollary}
%
It is known that the least upper semicontinuous majorant of 
a family of plurisubharmonic funtions is either a constant function
$+\infty$ or plurisubharmonic
(cf. Theorem 5 in \cite[Chapter II, \S2]{MR0243112}).
Kerckhoff's formula \eqref{eq:Kerckhof_formula}
implies that the Teichm\"uller distance $d_{T}(x_{0},\cdot)$ from a reference point $x_{0}\in \teich{g,m}$
coincides with the half of the least upper semicontinuous majorant
of the family of log-extremal length functions.
Therefore,
we obtain an alternative proof of Krushkal's result
\cite[Corollary 3]{MR1142683}
as follows.

\begin{corollary}[Teichm\"uller distance is plurisubharmonic]
\label{coro:Teichmuller distance}
For $x_{0}\in \teich{g,m}$,
the distance function
$$
\teich{g,m}\ni x\mapsto d_{T}(x_{0},x)
$$
is plurisubharmonic.
\end{corollary}

\subsection{Convexities for Teichm\"uller space}
A subset $K$ in a complex manifold $N$ is said to be
\emph{disk-convex} in $N$
if for any continuous mapping $g\colon \overline{\mathbb{D}}\to N$,
holomorphic in $\mathbb{D}$,
$f(\partial \mathbb{D})\subset K$ implies
$f(\overline{\mathbb{D}})\subset K$
(cf. \cite{MR1478844}).
For $\epsilon>0$ and a measured foliation $F$,
we call the set of the form $\{x\in \teich{g,m}\mid \ext_{x}(F)<\epsilon\}$
the \emph{$\epsilon$-horoball for $F$}.
From the maximum modulus principle
for (pluri)subharmonic functions,
we obtain the following.

\begin{corollary}[Disk convexity]
\label{coro:diskconvexity}
For $\epsilon>0$ and a measured foliation $F$,
the $\epsilon$-horoball for $F$ is disk-convex  in $\teich{g,m}$.
The metric ball with respect to the Teichm\"uller distance
is also disk-convex  in $\teich{g,m}$.
\end{corollary}

About the convexity of metric balls,
A. Lenzhen and K. Rafi \cite{MR2838267} observed that
the metric ball with respect to the Teichm\"uller distance
is quasi-convex.
Furthermore,
they also showed that the extremal length function is not convex along a Teichm\"uller geodesic
in general.
This means that the intersection between a horoball of extremal length and a Teichm\"uller disk
is simply connected,
but may not be hyperbolically convex in general.

A complex manifold $N$ is said to be \emph{hyperconvex}
(in the sense of Stehl\'e)
if it admits a continuous plurisubharmonic exhaustion function
$\rho\colon N\to [-\infty,0)$
in the sense that
$\{x\in N\mid \rho(x)<c\}$ is relatively compact in $M$ for every $c<0$ (cf. \cite{MR0374493}. See also \cite{MR1210002}).
%The Teichm\"uller space $\teich{g,m}$ is biholomorphically equivalent
%to a bounded domain in $\mathbb{C}^{3g-3+m}$
%(cf. \cite[Chapter 6]{MR1215481}).
Take two essentially complete measured foliations $F$ and $G$
%$F$ and $G$ which have only three prong singularities
%and one prong singularities at punctures
%and contain no saddle connections.
which are transverse in the sense that
\begin{equation}
\label{eq:transverse-mf}
i(F,H)+i(G,H)>0
\end{equation}
for all $H\in \mathcal{MF}-\{0\}$.
In Theorem \ref{thm:plurisuperharmonicitiy-of-reciprocals}
given later,
we will check that
\begin{equation}
\label{eq:extremal-length-exhausion}
\rho\colon \teich{g,m}\ni x\mapsto -\frac{1}{\ext_{x}(F)+\ext_{x}(G)+1}
\end{equation}
is a real analytic, negative, strictly plurisubharmonic exhaustion function
with lower bounds.
%(this is not immediate from Theorem \ref{thm:pluri-subharmonicity}).
Thus we obtain the following.
%
%(cf. \S\ref{subsec:hyperconvexity}).
\begin{corollary}[Hyperconvexitiy]
\label{coro:hyperconvexity}
Teichm\"uller space is hyperconvex.
\end{corollary}
Corollary \ref{coro:hyperconvexity}
is already given by Krushkal in \cite{MR1119946}.
Notice that the hyperconvexity of Teichm\"uller space
also follows from the compleness of the Carath\'eodory distance
(cf. \cite{MR534932} and \cite{MR0352450}).
%is a positive, proper, real analytic and strictly plurisubharmonic function
%(cf. \S\ref{Kerckhoff-Minsky}).
As an immediate corollary of the hyperconvexity
(or Corollary \ref{coro:Teichmuller distance}),
by virtue of Oka's theorem,
we deduce the following,
which is first proven by L.Bers and L.Ehrenpreis
(see \cite{MR0168800}.
See also \cite{MR0352450},
\cite{2012arXiv1210.0743L},
\cite{MR766636}, 
\cite{MR880186}).

\begin{corollary}[Holomorphic convexity]
\label{coro:stein}
Teichm\"uller space is
a domain of holomorphy.
\end{corollary}
%
%A subset $K$ in a complex manifold $N$ is said to be
%\emph{disk-convex} in $N$
%if for any continuous mapping $g\colon \overline{\mathbb{D}}\to N$,
%holomorphic in $\mathbb{D}$,
%$f(\partial \mathbb{D})\subset K$ implies
%$f(\overline{\mathbb{D}})\subset K$
%(cf. \cite{MR1478844}).
%For $\epsilon>0$ and a measured foliation $F$,
%we call the set of the form $\{x\in \teich{g,m}\mid \ext_{x}(F)<\epsilon\}$
%the \emph{$\epsilon$-horoball for $F$}.
%From the maximum modulus principle
%for (pluri)subharmonic functions,
%we obtain the following.
%
%\begin{corollary}[Disk convexity]
%\label{coro:diskconvexity}
%For $\epsilon>0$ and a measured foliation $F$,
%the $\epsilon$-horoball for $F$ is disk-convex  in $\teich{g,m}$.
%The metric ball with respect to the Teichm\"uller distance
%is also disk-convex  in $\teich{g,m}$.
%\end{corollary}
%
%About the convexity of metric balls,
%A. Lenzhen and K. Rafi \cite{MR2838267} observed that
%the metric ball with respect to the Teichm\"uller distance
%is quasi-convex.
%Furthermore,
%they also showed that the extremal length function is not convex along a Teichm\"uller geodesic
%in general.
%This means that the intersection between a horoball of extremal length and a Teichm\"uller disk
%is simply connected,
%but may not be hyperbolically convex in general.
%
%C. McMullen \cite{MR1478844}
%showed that the quasifuchsian space is disk-convex in the representation space.
%Since the quasifuchsian space is biholmorphically equivalent to
%the product of Teichm\"uller space,
%Corollary \ref{coro:Teichmuller distance} indicates 
%a qualitative
%improvement of McMullen's disk-convexity as follows.
%
%\begin{corollary}[Qualitative disk-convexity of $\mathcal{QF}$]
%\label{coro:qualitative-disk-convexity}
%Let $g$ be a continuous mapping on $\overline{\mathbb{D}}$
%which is holomorphic on $\mathbb{D}$.
%If $g(\mathbb{D})$ consists of $K$-quasifuchsian groups,
%then $g(\mathbb{D})$ also consists of $K$-quasifuchsian groups.
%\end{corollary}
%
%Where a quasifuchsian group is said to be a $K$-\emph{quasifuchsian group}
%if the limit set is $K$-quasicircle.
%Corollary \ref{coro:qualitative-disk-convexity} is not obvious
%as compared to the previous corollaries.
%Hence we shall give a proof in \S\ref{proof-of-disk-convexity}.

\section{Notation}

\subsection{Teichm\"uller space}
Let $\Sigma_{g,m}$ be a compact orientable surface of genus $g$
with $m$-disks removed with $2g-2+m>0$.
The \emph{Teichm\"uller space} $\teich{g,m}$ of
Riemann surfaces of analytically finite type $(g,m)$
is the set of equivalence classes of
pairs $(M,f)$ consisting of a Riemann surface $M$ of analytically finite type $(g,m)$
and an orientation preserving homeomorphism $f\colon {\rm Int}(\Sigma_{g,m})\to M$.
Two marked Riemann surfaces $(M_{1},f_{1})$ and $(M_{2},f_{2})$
are \emph{equivalent} if
there is a conformal mapping $h\colon M_{1}\to M_{2}$ which homotopic to $f_{2}\circ f_{1}^{-1}$.

The Teichm\"uller space admits a canonical distance $d_{T}$,
which we call the \emph{Teichm\"uller distance}.
The Teichm\"uller distance is originally 
defined as the logarithm of 
the infimum of the maximal dilatations
of quasiconformal mappings respecting the markings
(cf. \cite[\S5]{MR1215481}).
S. Kerckhoff \cite{MR559474}
gave a geometric description of the Teichm\"uller distance
via the extremal length,
which we will recall in \eqref{eq:Kerckhof_formula} below.

\subsection{Measured foliations}
\label{subsec:measured-foliatons}
Let $\mathcal{S}$ be the set of homotopy classes
of non-trivial and non-peripheral simple closed curves on $\Sigma_{g,m}$.
Let $\mathcal{WS}$ be the set of weighted simple closed curves,
that is, the set of formal products
$t\alpha$ of non-negative number $t$ and $\alpha\in \mathcal{S}$.
The closure $\mathcal{MF}$ of the embedding
$$
\mathcal{WS}\ni t\alpha
\mapsto [\mathcal{S}\ni \beta\mapsto t\,i(\alpha,\beta)]\in
\mathbb{R}_{+}^{\mathcal{S}}
$$
is called the \emph{space of measured foliations} on $\Sigma_{g,m}$.
We consider $\mathcal{WS}$ as
a subset of $\mathcal{MF}$.
We identify $1\cdot \alpha\in \mathcal{WS}$
with $\alpha\in \mathcal{S}$.
Any measured foliation is described as a pair consisting
of a singular foliation and a transverse measure
(cf. \cite[Expos\'e 5]{MR568308}).
For instance,
$t\alpha\in \mathcal{MF}$ is a foliated annulus with core $\alpha$
which identified with an annulus
$[0,t]\times [0,1]/(x,0)\sim (x,1)$
and a transverse measure associated to $|dx|$.

It is known that
the geometric intersection number $i(\alpha,\beta)$
for $\alpha,\beta\in \mathcal{S}$
extends continuously to $\mathcal{MF}\times \mathcal{MF}$
(cf. \cite{MR847953}).

If we fix a complete hyperbolic structure on the interior of $\Sigma_{g,m}$,
any measured foliation canonically corresponds to a measured lamination.
A \emph{measured lamination} is a pair of a geodesic lamination,
which called the \emph{support},
and a transverse measure,
where a \emph{geodesic lamination} a  is a compact set 
in the interior fo $\Sigma_{g,m}$ which consists
of disjoint complete geodesics
(cf. \cite[\S1.7]{MR1144770}).
A measured foliation is said to be \emph{essentially complete}
if the support of the corresponding measured lamination is maximal,
that is,
the complement consists of ideal triangles or ideal punctured monogon
if $(g,m)\ne (1,1)$,
a punctured bigon otherwise
(cf. \cite[Definition 9.5.1, Propositions 9.5.2 and 9.5.4]{Thuston-LectureNote}).
In the interior of $\Sigma_{g,m}$,
each singularity of the associated foliation of an essentially complete measured foliation is a three prong singularity.
At any puncture,
the associated foliation has a one prong singularity
(cf. \cite[Epilogue]{MR1144770}).

\subsection{Extremal length}
\label{subsec:extremal_length}
Let $M$ be a Riemann surface and let $A$ be a doubly connected domain
on $M$.
If $A$ is conformally equivalent to a round annulus $\{1<|z|<R\}$,
we define the \emph{modulus} of $A$ by
$$
\Mod(A)=\frac{1}{2\pi}\log R.
$$
The \emph{extremal length} of
a simple closed curve $\alpha$
on $M$ is defined by
\begin{equation}
\label{eq:gemetric_definition_extremal_length}
\ext_M(\alpha)=\inf\left\{\frac{1}{\Mod(A)}\mid
\mbox{the core curve of $A\subset M$ is homotopic to $\alpha$}
\right\}.
\end{equation}
In \cite{MR559474},
Kerckhoff showed that 
if we define the extremal length of $t\alpha\in \mathcal{WS}$
by
$$
\ext_M(t\alpha)=t^2\ext_M(\alpha),
$$
then the extremal length function $\ext_M$ on $\mathcal{WS}$
extends continuously to $\mathcal{MF}$.
For $F\in \mathcal{MF}$,
the extremal length function $\ext_{\cdot}(F)$
on $\teich{g,m}$ is defined by
$$
\ext_x(F)=\ext_M(f(F)).
$$
It is known that the function
$$
\teich{g,m}\times \mathcal{MF}\ni (x,F)\mapsto \ext_{x}(F)
$$
is continuous.
Furthermore,
the extremal length function $\ext_{x}\colon\mathcal{MF}\to \mathbb{R}$
is a proper function and satisfies the quasiconformal distortion property:
$$
\ext_{y}(F)\le e^{2d_{T}(x,y)}\ext_{x}(F)
$$
for any $F\in \mathcal{MF}$ and $x,y\in \teich{g,m}$.

%\medskip
%\noindent
%\paragraph{{\bf Kerckhoff's formula and Minsky's inequality}}
%As we noted in \S\ref{subsec:Background_Motivation},
\subsection{Kerckhoff's formula and Minsky's inequality}
\label{Kerckhoff-Minsky}
The Teichm\"uller distance $d_T$ is expressed with the extremal length,
which we call \emph{Kerckhoff's formula}:
\begin{align}
\label{eq:Kerckhof_formula}
d_T(x_1,x_2)&=\frac{1}{2}\log \sup_{\alpha\in \mathcal{S}}
\frac{\ext_{x_2}(\alpha)}{\ext_{x_1}(\alpha)}
\\
&=
\frac{1}{2}\sup_{\alpha\in \mathcal{S}}
\left(
\log\ext_{x_2}(\alpha)-\log\ext_{x_1}(\alpha)
\right)
\nonumber
\end{align}
for $x_{1},x_{2}\in \teich{g,m}$
(see \cite{MR559474}).
Y. Minsky \cite{MR1231968} observed the following inequality,
which we recently call \emph{Minsky's inequality}:
\begin{equation}
\label{eq:Minsky_inequality}
i(F,G)^2\le \ext_x(F)\,\ext_x(G)
\end{equation}
for $x\in \teich{g,m}$ and $F,G\in \mathcal{MF}$.
If two measured foliation $F$ and $G$ are transverse
in the sense of \eqref{eq:transverse-mf},
we have
$$
\ext_{x}(F)+\ext_{x}(G)\ge\frac{i(F,H)^{2}+i(G,H)^{2}}{\ext_{x}(H)}
$$
for all $H$ and hence we deduce
$$
\ext_{x}(F)+\ext_{x}(G)\ge e^{2d_{T}(x_{0},x)}
\min\{i(F,H)^{2}+i(G,H)^{2}\mid \ext_{x_{0}}(H)=1\}.
$$
This implies that the function $x\mapsto \ext_{x}(F)+\ext_{x}(G)$
is proper on $\teich{g,m}$.
%Since $\mathcal{PMF}$ is compact,
%for any $y_1,y_2\in \teich$,
%there are $F,G\in \mathcal{MF}$ such that
%$$
%e^{2d_T(y_1,y_2)}
%=\frac{\ext_{y_2}(F)}{\ext_{y_1}(F)}
%=\frac{\ext_{y_1}(G)}{\ext_{y_2}(G)}.
%$$
%In fact, such $F$ and $G$ are unique up to multiplying by positive constants,
%and are realized as the horizontal and vertical foliations
%of a holomorphic quadratic differential
%which defines a Teichm\"uller geodesic connecting $y_1$ and $y_2$
%(cf. \citeMR1099913 and \S\ref{subsec:Teichmuller_rays}).
%

%\medskip
%\noindent
%\paragraph{{\bf Minsky's inequality}}
%Minsky \cite{MR1231968} observed the following inequality,
%which we recently call \emph{Minsky's inequality}:
%\begin{equation}
%\label{eq:Minsky_inequality}
%i(F,G)^2\le \ext_y(F)\,\ext_y(G)
%\end{equation}
%for $y\in \teich{g,m}$ and $F,G\in \mathcal{MF}$.
%Minsky's inequality is sharp in the sense that
%for any $y\in \teich{g,m}$ and $F\in \mathcal{MF}$,
%there is a unique $G\in \mathcal{MF}$ up to multiplication by a positive constant such that
%$i(F,G)^2=\ext_y(F)\,\ext_y(G)$
%(cf. \citeMR1099913).

\subsection{Quadratic differentials}
Let $\quaddiff{g,m}$ be the space of holomorphic quadratic differentials
with finite $L^{1}$-norm.
Namely,
the space $\quaddiff{g,m}$ is a holomorphic vector bundle over $\teich{g,m}$
and the fiber $\quaddiff{x}$ at $x=(M,f)\in \teich{g,m}$ is
the space of holomorphic quadratic differentials on $M$
with finite $L^{1}$-norm
(cf. \S\ref{subsec:infinitesimal-theory}).
Any $q\in \quaddiff{x}$ extends meromorphically to
the completion $\overline{M}$ by filling punctures
and has (at most) simple poles at punctures.
A quadratic differential $q\in \quaddiff{g,m}$ is said to be \emph{generic}
if it satisfies the following two conditions:
\begin{enumerate}
\item
$q$ has a simple pole at every puncture.
\item
all zeroes of $q$ are simple.
\end{enumerate}
We can easily check that the set 
%$\quaddiff{g,m}^{0}$ of
of all generic holomorphic quadratic differentials is an open and dense subset in 
$\quaddiff{g,m}$.
%and the complement $\quaddiff{g,m}\setminus \quaddiff{g,m}^{0}$
%is analytic subvariety in $\quaddiff{g,m}$.


\subsection{Hubbard-Masur differentials}
For $q\in \quaddiff{g,m}$,
there is a unique measured foliation $F$ such that
$$
i(F,\alpha)=\inf_{\alpha'\sim f(\alpha)}\int_{\alpha'}|{\rm Re}\sqrt{q}|.
$$
for all $\alpha\in \mathcal{S}$,
where $(M,f)=\pi(q)\in \teich{g,m}$.
We call $F$ the \emph{vertical foliation} and denote it by
$\vertical(q)$.
We call $\horizontal(q)=\vertical(-q)$
the \emph{horizontal foliation} of $q$.
%For $F\in \mathcal{MF}$ and $x\in \teich{g,m},
%let $q_{F,x}\in \quaddiff{x}$ be the Hubbard-Masur differential for $F$ on $x$.
%Namely,
%the vertical foliation of $q_{F,x}$ is equivalent to $f(F)$ as measured foliations.
%For $\alpha\in \mathcal{S}$,
%we call $q_{\alpha,x}$ is the Jenkins-Strebel differential for $\alpha$ on $x$.
%
%
It is known that for any $F\in \mathcal{MF}$
and $x=(M,f)\in \teich{g,m}$,
there is a unique holomorphic quadratic differential
$q_{F,x}$ such that
%$$
%i(F,\alpha)=\inf_{\alpha'\sim f(\alpha)}\int_{\alpha'}|{\rm Re}\sqrt{q_{F,x}}|.
%$$
%Namely,
%the \emph{vertical foliation} of $q_{F,x}$ is equivalent to $F$.
$\vertical(q_{F,x})=F$.
We call $q_{F,x}$ the \emph{Hubbard-Masur differential}
for $F$ on $x$
(cf. \cite{MR523212}).
If $F$ is essentially complete,
the Hubbard-Masur differential $q_{F,x}$ is generic for all $x\in \teich{g,m}$,
since any Whitehead equivalent measured foliation to $F$
is isotopic to $F$.
%that is,
%there are no collapsing or saddle connections
%in the transforming procedure.

The Hubbard-Masur differential $q_{F,x}=q_{F,x}(z)dz^2$
for $F$ on $x=(M,f)$ satisfies
\begin{equation}
\label{eq:JS_extremallength_norm_length}
\ext_x(F)=\|q_{F,x}\|=\int_M|q_{F,x}|=\int_M|q_{F,x}(z)|
\cdot\Leb{z}.
\end{equation}

%where $\Leb{z}=\frac{i}{2}dz\wedge d\overline{z}$ is the standard Lebesgue meaure
%on $\mathbb{C}\cong \mathbb{R}^{2}$.
%In particular,
%it is known that
%\begin{equation}
%\label{eq:JS_extremallength_norm_length}
%\ext_y(\alpha)=\|q_{\alpha,y}\|=\frac{\ell_{q_{G,y}}(\alpha)^2}{\|q_{\alpha,y}\|}
%\end{equation}
%where $\ell_{q_{G,y}}(\alpha)$ is the length of the geodesic representative
%homotopic to $f(\alpha)$ with respect to the singular flat metric
%$|q_{\alpha,y}|=|q_{\alpha,y}(z)||dz|^2$.


\subsection{Infinitesimal theory}
\label{subsec:infinitesimal-theory}
Teichm\"uller space also has a canonical complex structure,
which induced from the deformation by quasiconformal mappings.
A \emph{Beltrami differential}
is,
by definition,
an $L^{\infty}$-measurable $(-1,1)$-form.
The holomorphic tangent space
is described as the quotient of the complex Banach space
of the Beltami differentials
(cf. \cite[Theorem 7.5]{MR1215481}).
The holomorphic cotangent space is identified with the space
of holomorphic quadratic differentials.
The natural pairing between holomorphic tangent space
and cotangent space is given by
$$
\langle v,q\rangle=\int_{M}\dot{\mu}q=
\int_{M}\dot{\mu}(z)q(z)\cdot\Leb{z}
$$
for $v=[\dot{\mu}]\in T_{x}\teich{g,m}$
and $q\in T^{*}_{x}\teich{g,m}\cong \quaddiff{x}$
and $x=(M,f)\in \teich{g,m}$.

\begin{convention}
Henceforth,
for $(p_{i},q_{i})$ forms $\varphi_{i}=\varphi_{i}(z)dz^{p_{i}}d\overline{z}^{q_{i}}$ $(i=1,2)$ on a Riemann surface $M$
with $p_{1}+p_{2}=q_{1}+q_{2}=1$,
we write
$$
\int_{M}\varphi_{1}\varphi_{2}=
\int_{M}\varphi_{1}(z)\varphi_{2}(z)\cdot\Leb{z}.
$$
\end{convention}


\section{Coordinates via representations of the odd cohomology}
%Applying this observation to the vector bundle
%$\pi\colon \quaddiff{g,m}\to \teich{g,m}$,
%we deduce the following.
%
%\begin{lemma}
%\label{lem:transversal}
%Let $F\colon \mathbb{D}\to \quaddiff{g,m}$ be a $C^{k+1}$-mapping
%such that $g(\lambda)=\pi\circ F(\lambda)\colon \mathbb{D}\to \teich{g,m}$ is holomorphic.
%Then,
%$\eta_{\lambda}=\left.F_{*}\left(\frac{\partial}{\partial \overline{\lambda}}
%\right)
%\right|_{\lambda}$
%is canonically identified with an element in $\quaddiff{g(\lambda)}$,
%and
%$\{\eta_{\lambda}\}_{\lambda\in \mathbb{D}}$
%is a $C^{k}$-family of holomorphic quadratic differentials
%over $\mathbb{D}$.
%\end{lemma}


\subsection{The tangent spaces to quadratic differentials}
\label{subsec:tangent-space-to-quadratic-differentials}
Following \cite{MR523212},
we describe the tangent space $T_{q_{0}}\quaddiff{g,m}$
as the first hypercohomolgy group $\mathbb{H}^{1}(L^{\bullet})$
a complex of sheaves
(cf. \cite{MR0102797} or \cite{MR507725}).
We will need the Kodaira-Spencer identification of the
tangent space of Teichm\"uller space
with the first cohomology group of the sheaf of holomorphic vector fields
(for instance, see \cite{MR815922}. See also \cite{MR1215481} and \cite{MR499279}).

Let $X$ and $q$
be a holomorphic vector field 
and a holomorphic quadratic differential
on an open set of a Riemann surface $M$.
Denote by $L_{X}q$ the Lie derivative of $q$ along $X$.
Let $\Theta_{M}$ and $\Omega_{M}^{\otimes 2}$ be the sheaves of germs of
holomorphic vector fields with zeroes at punctures
and meromorphic quadratic differentials on $M$ with (at most)
first order poles at punctures,
respectively.
Let $q_{0}\in \quaddiff{g,m}$
($q_{0}$ need not to be generic).
The tangent space $T_{q_{0}}\quaddiff{g,m}$
is identified with the 
first hypercohomology group of the complex of sheaves
$$
\begin{CD}
L^{\bullet}\colon\quad
0 @>>> \Theta_{M} @>{L_{\cdot}q_{0}}>> \Omega_{M}^{\otimes 2}@>>>0
\end{CD}
$$
(cf. \cite[Proposition 4.5]{MR523212}).
For the convenience,
we recall the definition of the (first)
hypercohomology group which we use here.
The first cochain group is the direct sum
$C^{0}(M,\Omega_{M}^{\otimes 2})\oplus C^{1}(M,\Theta_{M})$.
Consider an appropriate covering $\mathcal{U}=\{U_{i}\}_{i}$ on $M$
such that $\mathbb{H}^{1}(L^{\bullet})\cong \mathbb{H}^{1}(\mathcal{U},L^{\bullet})$
(see the proof of \cite[Proposition 4.5]{MR523212}).
A cochain $(\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})$
in $C^{0}(\mathcal{U},\Omega_{M}^{\otimes 2})\oplus
C^{1}(\mathcal{U},\Theta_{M})$
is said to be \emph{cocycle} if
it satisfies
$$
\delta\{X_{ij}\}_{i,j}=X_{ij}+X_{jk}+X_{ki}=0,\quad
\delta\{\phi_{i}\}_{i}=\phi_{i}-\phi_{j}=L_{X_{ij}}(q_{0}).
$$
A \emph{coboundary} is a cochain $(\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})$
of the form
$$
X_{ij}=Z_{i}-Z_{j}=\delta\{Z_{i}\}_{i},\quad
\phi_{i}=L_{Z_{i}}(q_{0})
$$
for some $0$-cochain $\{Z_{i}\}_{i}\in C^{0}(\mathcal{U},\Theta_{M})$
(cf. Figure \ref{fig:hypercohomology}).
%\begin{equation}
%\label{eq:hypercohomology}
\begin{figure}[t]
$$
\begin{CD}
0
\\
@AAA \\
C^{0}(\mathcal{U},\Omega_{M}^{\otimes 2})
@>{\delta}>>
C^{1}(\mathcal{U},\Omega_{M}^{\otimes 2}) \\
@A{L_{\cdot}q_{0}}AA
@A{-L_{\cdot}q_{0}}AA
\\
C^{0}(\mathcal{U},\Theta_{M})
@>{\delta}>>
C^{1}(\mathcal{U},\Theta_{M})
@>{\delta}>>
C^{2}(\mathcal{U},\Theta_{M})
%\\
%@AAA @AAA @AAA
%\\
%0 @. 0 @. 0
\end{CD}
$$
%
%$$
%\begin{CD}
%0@>>> C^{2}(\mathcal{U},\Theta_{M}) \\
%@. @A{\delta}AA  \\
%0@>>> C^{1}(\mathcal{U},\Theta_{M}) @>{L_{\cdot}q_{0}}>> 
%C^{1}(\mathcal{U},\Omega_{M}^{\otimes 2}) \\
%@. @A{\delta}AA @A{-\delta}AA \\
%0@>>> C^{0}(\mathcal{U},\Theta_{M}) @>{L_{\cdot}q_{0}}>> C^{0}(\mathcal{U},\Omega_{M}^{\otimes 2}) @>>> 0\\
%\end{CD}
%%\end{equation}
%$$
%$$
%\begin{CD}
%0@>>> C^{2}(\mathcal{U},\Theta_{M}) \\
%@. @A{\delta}AA  \\
%0@>>> C^{1}(\mathcal{U},\Theta_{M}) @>{L_{\cdot}q_{0}}>> C^{1}(\mathcal{U},\Omega_{M}^{\otimes 2}) \\
%@. @A{\delta}AA @A{-\delta}AA \\
%0@>>> C^{0}(\mathcal{U},\Theta_{M}) @>{L_{\cdot}q_{0}}>> C^{0}(\mathcal{U},\Omega_{M}^{\otimes 2}) @>>> 0\\
%\end{CD}
%%\end{equation}
%$$
\caption{Double complex for the tangent spaces to $\quaddiff{g,m}$}
\label{fig:hypercohomology}
\end{figure}

%Notice that for any cocycle $\{X_{ij}\}_{ij}\in C^{1}(M,\Theta_{M})$,
%we can always find a $0$-cochain
%$\{X_{i}\}_{i}$ of the sheaf
%of $C^{\infty}$-vector fields which satisfies
%$X_{i}-X_{j}=X_{ij}$ on $U_{i}\cap U_{j}$
%applying a partition of unity.
%Then,
%the $\overline{z}$-derivative
%$\dot{\mu}=(X_{i})_{\overline{z}}$
%defines a Beltrami differential on $M$
%which
%represents the infinitesimal deformation
%corresponding
%to the tangent vector
%associated to
%the cohomology class of $\{X_{ij}\}_{i,j}$ in $H^{1}(M,\Theta_{M})$.
%%that is the tangent space of $\teich{g,m}$ at $M$.
%See \cite[\S7.2.4]{MR1215481}.{\color{red}これは
%$\dot{\mu}=-(X_{i})_{\overline{z}}$
%ではなかろうか？}
%090-9643-2443
%03-3592-0985
%
%\begin{lemma}
%Let $M$ be a Riemann surface
%and $\mathcal{U}=\{U_{i}\}_{i}$ be a covering of $M$
%such that the intersection of any finite subcollection in $\mathcal{U}$
%is connected and simply connected.
%Let $(\{X_{ij}\}_{i,j},\{\phi_{i}\}_{i})$ be a cocycle
%in $C^{1}(\mathcal{U},\Theta_{M})\oplus C^{0}(\mathcal{U},\Omega_{M}^{\otimes 2})$.
%Suppose that $q_{0}$ is the square of a holomorphic $1$-form $\omega$ on $M$,
%and $\omega=\omega_{i}(z_{i})dz_{i}$ on $U_{i}$.
%Suppose also that $\phi_{i}/\omega_{i}$ is holomorphic on $U_{i}$ for each $i$.
%Then, family of $1$-forms
%$$
%\Omega_{i}=\left(
%\frac{\phi_{i}}{\omega_{i}}-2\omega_{i}'X_{i}-2\omega_{i}(X_{i})_{z}
%\right)dz_{i}
%-(2\omega_{i}(X_{i})_{\overline{z}}
%)d\overline{z}_{i}
%$$
%on $U_{i}$ defines a closed $1$-form $\Omega$ on $M$.
%\end{lemma}
%
%\begin{proof}
%The closedness of $\Omega_{i}$ follows from the calculation.
%We need to check that $\Omega_{i}$ is well-defined.
%Indeed,
%on $U_{i}\cap U_{j}$,
%\begin{align*}
%\frac{\phi_{i}}{\alpha_{i}}
%-\frac{\phi_{j}}{\alpha_{j}}
%\end{align*}
%
%
%
%\end{proof}


\subsection{Branched covering space associated with generic differentials}
\label{subsec:Branched-Covering-Space-associated-with-generic-differentials}
Let $x=(M,f)\in \teich{g,m}$ and $q_{0}$ a generic holomorphic
quadratic differential on $M$.
We consider $M$ as a pair of topological surface $\Sigma_{g}$
and a complex strtucture on $\Sigma_{g}-\{\mbox{punctures of $M$}\}$.

Let $U$ be a contractible neighborhood of $q_{0}$ in $\quaddiff{g,m}^{0}$.
For $q\in U$,
let $A_{q}={\rm Zero}(q)\cup {\rm Pole}(q)\subset \Sigma_{g}$
and $M_{q}$ the underlying surface of $q$.
Then,
we have a homomorphism
\begin{equation}
\label{eq:representation-q}
\rho_{q}\colon \pi_{1}(\Sigma_{g}-A_{q})\to \mathbb{Z}/2\mathbb{Z}
\subset {\rm Isom}(\mathbb{C})
\end{equation}
associated to the double covering space
$$
\pi_{q}\colon \tilde{M}_{q}\to M_{q}
$$
of Riemann surface of $\sqrt{q}$
over $\Sigma_{g}-A_{q}$,
where $\mathbb{Z}/2\mathbb{Z}$ in \eqref{eq:representation-q}
is recognized as
a subgroup of the isometry group ${\rm Isom}(\mathbb{C})$
of $\mathbb{C}$ with the standard Euclidean metric
generated by the $\pi$-rotation.
The square root $\sqrt{q}$ lifts to a well-defined holomorphic $1$-
form $\omega_{q}$ on $\tilde{M}_{q}$.
%For simplicity of notation,
%we set $\rho_{0}=\rho_{q_{0}}$ and $A_{0}=A_{q_{0}}$.

Let $W=\Sigma_{g}-A_{q_{0}}$
and $W'$ be a compact subsurface of $W$
with smooth boundary
such that
the inclusion $W'\hookrightarrow W_{0}$
is homotopy equivalent and
the complement $\Sigma_{g}-W'$
is a union of tiny open $|q_{0}|$-disks of centers in $A_{0}$
By taking sufficiently small $U$ if necessary,
we may assume that $A_{q}\subset  \Sigma_{g}-W'$
for any $q\in U$.
Hence,
by passing the inclusions $W'\hookrightarrow W$
and $W'\hookrightarrow \Sigma_{g}-A_{q}$,
we get a canonical identification between
$\pi_{1}(W)$ and $\pi_{1}(\Sigma_{g}-A_{q})$
so that $\rho_{q}=\rho_{0}$ as a homomorphism from
$\pi_{1}(W)$ to $\mathbb{Z}/2\mathbb{Z}$
for all $q\in U$.
%there is a homeomorphism $h_{q}\colon \sigma_{g}\to \Sigma_{g}$ isotopic to the identity
%such that $h_{q}(A_{q_{0}})=A_{q}$,
%$h_{q_{0}}=id_{\Sigma_{g}}$ and
%the mapping
%$$
%U\times \Sigma_{g}\ni (q,x)\to h_{q}(x)\in \Sigma_{g}
%$$
%is continuous.
%Then,
%we can see easily that $\rho_{q}\circ (h_{q})_{*}=\rho_{q_{0}}$.
%Let $\mathbb{C}_{\rho_{0}}$ be 
%the local coefficient system for an abelian group $\mathbb{C}$ over $W$
%associated to $\rho_{0}$.
%
%Let $\tilde{W}\to\Sigma_{g}$ be a branched
%double covering space associated to the kernel of $\rho_{0}$
%(branched over points in $A_{0}$).
%Let $\tilde{A}_{0}\subset \tilde{W}$ be the set of branced points.
%The genus of $\tilde{W}$ is $6g-5+2m$
%and the number of $\tilde{A}_{0}$ is $4g-4+2m$.
%Let us denote by $\oddcohomology$
%the first odd cohomology group
%$H^{1}(\tilde{W},\tilde{A}_{0};\mathbb{C})$
%of the pair $(\tilde{W},\tilde{A}_{0})$ with complex coefficients.
%%$H^{1}(\tilde{W},\partial \tilde{W};\mathbb{C})^{-}$,
%Namely,
%it is the eigenspace of the cohomology group
%associated with the eigenvalue $-1$
%of the action of the covering transformation of $\tilde{W}\to \Sigma_{g}$.
%
%
%
%%Then,
%%we define the cohomology group $H^{1}(W,\partial W;\mathbb{R}^{2}_{\rho_{0}})$.
%
%\begin{lemma}[cf. Proposition 1.2 in \cite{MR1135877}]
%\label{lem:dimension}
%the complex dimension of the odd cohomology group $\oddcohomology$
%is equal to $6g-6+2m$.
%\end{lemma}
%
%\begin{proof}
%Let $\tilde{W}\to W$ be a double covering space associated to the kernel of $\rho_{0}$.
%The genus of $\tilde{W}$ is $6g-5+2m$
%and the number of the components of $\partial \tilde{W}$ is $4g-4+2m$.
%Then,
%there is a natural isomorphism
%$$
%H^{1}(W,\partial W;\mathbb{C}_{\rho_{0}})\cong 
%\oddcohomology
%$$
%where $H^{1}(W,\partial W;\mathbb{C}_{\rho_{0}})^{-}$
%is the odd part in $H^{1}(W,\partial W;\mathbb{C}_{\rho_{0}})$,
%that is,
%$$
%\oddcohomology
%=\{\xi\in H^{1}(\tilde{W},\partial \tilde{W};\mathbb{C})\mid T^{*}(\xi)=-\xi\}
%$$
%%that is,
%%the eigenspace associated to the eigenvalue $-1$
%%of the isomorphism $H^{1}(\tilde{W},\partial \tilde{W};\mathbb{C})$
%where $T$ is the covering transformation on $\tilde{W}$.
%Poncar\'e-Lefchetz duality theorem asserts that
%\begin{align*}
%\dim_{\mathbb{C}}H^{1}(\tilde{W},\partial \tilde{W};\mathbb{C})
%&=\dim_{\mathbb{C}}H_{1}(\tilde{W})
%=(6g-5+2m)+(4g-4+2m)-1\\
%&=10g-10+4m
%\end{align*}
%
%
%
%
%
%
%
%since $|{\rm Sing}(q)|=4g-4+2m${\color{red}(同じ計算?)}.
%\end{proof}

\subsection{Backgrounds}
\label{subsec:background-tangent-space}
%Let $X$ and $q$
%be a holomorphic vector field 
%and a holomorphic quadratic differential
%on an open set of $M_{q_{0}}$.
%Denote by $L_{X}q$ the Lie derivative of $q$ along $X$.
%Let $\Theta_{M_{q_{0}}}$ and $\Omega_{M_{q_{0}}}^{\otimes 2}$ be the sheaves of germs of
%holomorphic vector fields and holomorphic quadratic differentials on $X$,
%respectively.
%Then,
%the tangent space $T_{q_{0}}\quaddiff{g,m}$
%is identified with the 
%first hypercohomology group $\mathbb{H}^{1}(L^{\bullet})$
%of the complex of sheaves
%$$
%\begin{CD}
%0 @>>> \Theta_{M_{q_{0}}} @>{L_{\cdot}q_{0}}>> \Omega_{M_{q_{0}}}^{\otimes 2}@>>>0
%\end{CD}
%$$
%(cf. \cite[Proposition 4.5]{MR523212}).
Let $q_{0}\in \quaddiff{g,m}$ be a generic differential on $M_{q_{0}}$.
Take a neighborhood $U$ of $q_{0}$ as the previous section.
%Let
%$\pi_{q}\colon\tilde{M}_{q}\to M_{q}$ for $q\in U$
%be the double branched covering associated to $\sqrt{q}$,
%and $\omega_{q}$
%a holomorphic $1$-form on $\tilde{M}_{q}$.
%triangulations $\tilde{\Delta}_{q}$ and $\Delta_{q}$
%on $\tilde{M}_{q}$ and $M_{q}$
%as \S\ref{sec:local-coordinate}.
Then, $\mathbb{M}=\cup_{q\in U}\tilde{M}_{q}$ is a holomorphic family of closed Riemann surfaces over $U$,
and the arrangement of the covering transformation $r_{q}\colon
\tilde{M}_{q}\to \tilde{M}_{q}$ defines the holomorphic automorphism
${\bf r}$ of  $\mathbb{M}$ of order $2$.
Taking a sufficiently small $U$ if necessary,
we may assume that there is a covering $\{\mathcal{V}_{i}\}_{i}$
of $\mathbb{M}$ such that
\begin{enumerate}
\item
each $\mathcal{V}_{i}$ admits a biholomorphic mapping $\mathcal{A}_{i}$
onto the product domain $\mathbb{D}\times U$
such that its restriction to $\mathcal{V}_{i,q}=\mathcal{V}_{i}\cap \tilde{M}_{q}$ is a biholomorphism onto $\mathbb{D}\times \{q\}$
for each $q\in U$,
%where $V_{i}\subset \tilde{M}_{q_{0}}$ is a simply connected domain,
%\item
%$\mathcal{V}_{i}\cap \mathcal{V}_{j}$ is biholomorphic to
%$(V_{i}\cap V_{j})\times U$ via $\mathcal{A}_{i}$.
\item
for any finite $i_{1},\cdots,i_{k}$ and $q\in U$,
the intersection $\mathcal{V}_{i_{1},q}\cap \cdots\cap \mathcal{V}_{i_{k},q}$
is connected and simply connected,
\item
eacn $V_{i,q}$ contains at most one branch point of $\tilde{M}_{q}\to M_{q}$
and any branch point of $\tilde{M}_{q}\to M_{q}$ is contained in at most
one open set in $\{V_{i,q}\}_{i}$,
and
\item
the covering $\{\mathcal{V}_{i}\}_{i}$ is equivariant under ${\bf r}$
in the sense that 
$\{{\bf r}(\mathcal{V}_{i})\}_{i}=\{\mathcal{V}_{i}\}_{i}$
and $\mathcal{A}_{j}\circ {\bf r}=\mathcal{A}_{i}$
when ${\bf r}(\mathcal{V}_{i})=\mathcal{V}_{j}$.
\end{enumerate}
Set $D=\{\lambda=s+it\mid |s|,|t|<\delta_{0}\}$.
Let $\{q_{\lambda}\}_{\lambda\in D}$
be a differentiable family of holomorphic quadratic differenitals
with $q_{0}=q_{0}$ and $q_{\lambda}\in U$ for $\lambda\in D$.
For any $\lambda\in D$,
we let $\alpha_{i}(\lambda)\colon \mathcal{V}_{i,q_{\lambda}}\to \mathbb{D}$
by $\alpha_{i}(\lambda)=\mathcal{A}_{i}\mid_{\tilde{M}_{q_{\lambda}}}$
and set
$V_{\lambda;ij}=\alpha_{j}(\lambda)(\mathcal{V}_{i,q_{\lambda}}\cap \mathcal{V}_{j,q_{\lambda}})$,
and
$\alpha_{ij}(\lambda)=\alpha_{j}(\lambda)\circ \alpha_{i}(\lambda)^{-1}\colon V_{\lambda;ji}\to V_{\lambda;ij}$.
For simplicity,
set $\alpha_{i}=\alpha_{0;i}$,
$\alpha_{ij}=\alpha_{0;ij}$
and $V_{ij}=V_{0;ij}$.
Consider the image of $\alpha_{j}$ is in the $z_{j}$-plane.

Let $\tilde{q}_{\lambda;i}dz_{i}^{2}$ be the representation of $(\pi_{q_{\lambda}})^{*}(q_{\lambda})$
under the coordinate $\alpha_{i}(\lambda)$.
Let
\begin{align}
&
\tilde{\phi}_{\lambda;i}=\frac{\partial \tilde{q}_{\lambda;i}}{\partial \lambda},
\quad
%\psi_{\lambda;i}=\frac{\partial q_{\lambda;i}}{\partial \overline{\lambda}},\\
%&
\tilde{X}_{\lambda;i}=-\frac{\partial \alpha_{i}(\lambda)}{\partial \lambda}\circ
\alpha_{i}(\lambda)^{-1},
\label{eq:vector-fields}
%\quad
%Y_{\lambda;i}=-\frac{\partial \alpha_{i}(\lambda)}{\partial \overline{\lambda}}\circ
%\alpha_{i}(\lambda)^{-1}
\end{align}
on $\mathbb{D}$
for all $i$ and set
$$
\tilde{X}_{\lambda;ij}(z_{j})=\frac{\partial \alpha_{ij}(\lambda)}{\partial \lambda}(z_{i})
,
%\quad
%Y_{\lambda;ij}(z_{j})=\frac{\partial \alpha_{ij}(\lambda)}{\partial \overline{\lambda}}(z_{i})
$$
where $z_{j}=\alpha_{ij}(\lambda)(z_{i})$ and $z_{i}\in V_{\lambda;ij}$
(tildes mean objects (differentials or vector fields etc.)
on the covering space $\tilde{M}_{q_{0}}$
which obtained as lifts of objects on $M_{q_{0}}$).
Then,
$$
\tilde{X}_{\lambda;i}(z_{i})\frac{\partial}{\partial z_{i}},\quad
%Y_{\lambda;i}(z_{i})\frac{\partial}{\partial z_{i}},\quad
\tilde{X}_{\lambda;ij}(z_{j})\frac{\partial}{\partial z_{j}}
$$
are vector fields on appropriate sets of $\tilde{M}_{q_{\lambda}}$.
Set $\tilde{q}_{i}=\tilde{q}_{0;i}$
and
$\tilde{\phi}_{i}=\tilde{\phi}_{0;i}$.
%and $\psi_{i}=\psi_{0;i}$.
Notice that
\begin{equation}
\label{eq:subsec:background-tangent-space1}
\alpha_{ij}(\lambda)_{*}(\tilde{X}_{\lambda;i})-\tilde{X}_{\lambda;j}=\tilde{X}_{\lambda;ij},
%\quad
%\alpha_{ij}(\lambda)_{*}(Y_{\lambda;i})-Y_{\lambda;j}=Y_{\lambda;ij}
\end{equation}
%on $V_{\lambda;ji}$,
%and
\begin{align}
\alpha_{ij}(\lambda)^{*}(\tilde{\phi}_{\lambda;i})
-\tilde{\phi}_{\lambda;j}
&=L_{\tilde{X}_{\lambda;ij}}(\tilde{q}_{\lambda;j})
\label{eq:subsec:background-tangent-space2}
%\\
%\alpha_{ij}(\lambda)^{*}(\psi_{\lambda;i})
%-\psi_{\lambda;j}
%&=L_{Y_{\lambda;ij}}(q_{\lambda;j})
%\label{eq:subsec:background-tangent-space3}
\end{align}
on $V_{\lambda;ij}$.
We abbreviate these equations to
\begin{align}
&\tilde{X}_{\lambda;i}-\tilde{X}_{\lambda;j}=\tilde{X}_{\lambda;ij}
,\quad
%Y_{\lambda;i}-Y_{\lambda;j}=Y_{\lambda;ij} 
\tilde{\phi}_{\lambda;i}
-\tilde{\phi}_{\lambda;j}
=L_{\tilde{X}_{\lambda;ij}}(\tilde{q}_{\lambda}).
\label{eq:subsec:background-tangent-space4}
%\quad
%\psi_{\lambda;i}
%-\psi_{\lambda;j}
%=L_{Y_{\lambda;ij}}(q_{\lambda})
%\label{eq:subsec:background-tangent-space5}
\end{align}
for instance.
Notice again that $\omega_{q_{\lambda}}^{2}=(\pi_{q_{\lambda}})^{*}(q_{\lambda})$
for all $\lambda\in D$.

Notice that for any cocycle $\{X_{ij}\}_{ij}\in C^{1}(M,\Theta_{M})$,
we can always find a $0$-cochain
$\{X_{i}\}_{i}$ of the sheaf
of $C^{\infty}$-vector fields which satisfies
$X_{i}-X_{j}=X_{ij}$ on $U_{i}\cap U_{j}$
applying a partition of unity.
Then,
from \eqref{eq:vector-fields},
the $\overline{z}$-derivative
\begin{equation}
\label{eq:infinitesimal1}
\dot{\nu}=-(\tilde{X}_{i})_{\overline{z}}
\end{equation}
defines a Beltrami differential on $M$
which
represents the infinitesimal deformation
corresponding
to the tangent vector
associated to
the cohomology class of $\{X_{ij}\}_{i,j}$ in $H^{1}(M,\Theta_{M})$.
%that is the tangent space of $\teich{g,m}$ at $M$.
Compare with Equation $(7.27)$ in \cite[\S7.2.4]{MR1215481}.
%{\color{red}これは
%$\dot{\mu}=-(X_{i})_{\overline{z}}$
%ではなかろうか？}



\subsection{Local coordinates via homomorphisms}
Let $H_{1}(\tilde{M}_{q})^{-}$ is the odd part of the homology
of $\tilde{M}_{q}$ with coefficient in $\mathbb{Z}$.
Namely,
$H_{1}(\tilde{M}_{q})^{-}$ is the the eigen space of the eigen value $-1$
of the linear automorphism $(r_{q})_{*}$ on the first cohomology group 
of $\tilde{M}_{q}$ defined by the covering transformation $r_{q}$.
In \cite{MR0396936},
Douady and Hubbard consider the following mapping
\begin{equation}
\label{eq:Douady-Hubbard}
\douadyhubbard\colon
U\ni \lambda \mapsto \chi_{q}\in {\rm Hom}(H_{1}(\tilde{M}_{q_{0}})^{-},\mathbb{C})
\end{equation}
defined by
$$
\chi_{q}(c)=\int_{c}\omega_{q}
$$
where we identify $H_{1}(\tilde{M}_{q_{0}})^{-}\cong H_{1}(\tilde{M}_{q})^{-}$
in a canonical way as discussed in \S\ref{subsec:Branched-Covering-Space-associated-with-generic-differentials}.
Notice that
${\rm Hom}(H_{1}(\tilde{M}_{q_{0}})^{-},\mathbb{C})$ is a $\mathbb{C}$-vector space
of complex dimension $6g-6+2m$.
%Indeed,
%they calculate the derivative along the fiber of $\quaddiff{g,m}\to \teich{g,m}$.
We shall check the following.

\begin{lemma}[Differential of $\douadyhubbard$]
\label{lem:Douady-Hubbard}
Let $q_{0}\in \quaddiff{g,m}$ be a generic differential.
Let $v\in T_{q_{0}}\quaddiff{g,m}$ be a tangent vector
associated to a $1$-cochain $(\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})$.
Let $\{X_{i}\}_{i}$ be a $0$-cochain of the sheaf
of $C^{\infty}$-vector fields on $X_{q_{0}}$.
Then,
the derivative of the mapping \eqref{eq:Douady-Hubbard} along $v$
satisfies
\begin{equation}
\label{eq:lem:Douady-Hubbard1}
\douadyhubbard_{*}[v](c)
=\int_{c}
\Phi
%\Phi_{\lambda}
%,\quad
%\douadyhubbard(q_{\lambda})_{*}\left(\frac{\partial}{\partial \overline{\lambda}}\right)(c)
%=\int_{c}\Psi_{\lambda}
\end{equation}
for all $c\in H_{1}(\tilde{M}_{q_{0}})^{-}$,
where
$$
\Phi=\left(
\frac{\tilde{\phi}_{i}}{2\omega_{q_{0}}}-\omega_{q_{0}}'\tilde{X}_{i}-\omega_{q_{0}}(\tilde{X}_{i})_{z}
\right)dz
-\omega_{q_{0}}
(\tilde{X}_{i})_{\overline{z}}d\overline{z}
$$
on $\mathcal{V}_{i,0}$.
%and $\tilde{\phi}_{i}$
%and $\tilde{X}_{i}$ are lifts of $\phi_{i}$
%and $X_{i}$
%to $\tilde{M}_{q_{0}}$,
%respectively.
%where
%\begin{align*}
%\Phi&=
%(\tilde{\phi_{i}}-2\omega_{q_{0}}'\tilde{X}_{i}-2\omega_{q_{0}}(\tilde{X}_{i})_{z})dz
%-2\omega_{q_{\lambda}}
%(\tilde{X}_{i})_{\overline{z}}d\overline{z} \\
%\Psi&=
%(\tilde{\psi_{i}}-2\omega_{q_{0}}'\tilde{Y}_{i}-2\omega_{q_{0}}(\tilde{Y}_{i})_{z})dz
%-2\omega_{q_{\lambda}}
%(\tilde{Y}_{i})_{\overline{z}}d\overline{z} 
%\end{align*}
%on $\mathcal{V}_{i,q_{\lambda}}$.
\end{lemma}
%\begin{lemma}
%\label{lem:Douady-Hubbard}
%The derivative of the mapping \eqref{eq:Douady-Hubbard}
%satisfies
%\begin{equation}
%\label{eq:lem:Douady-Hubbard1}
%\douadyhubbard(q_{\lambda})_{*}\left(\frac{\partial}{\partial \lambda}\right)(c)
%=\int_{c}\Phi_{\lambda},\quad
%\douadyhubbard(q_{\lambda})_{*}\left(\frac{\partial}{\partial \overline{\lambda}}\right)(c)
%=\int_{c}\Psi_{\lambda}
%\end{equation}
%for all $c\in H_{1}(\tilde{M}_{q_{0}})^{-}$,
%where
%\begin{align*}
%\Phi_{\lambda}&=\frac{(\phi_{\lambda;i}-(q_{\lambda;i})'X_{\lambda;i}-2q_{\lambda;i}(X_{\lambda;i})_{z}}{\omega_{q_{\lambda}}}dz
%-2\omega_{q_{\lambda}}
%(X_{\lambda;i})_{\overline{z}}d\overline{z} \\
%\Psi_{\lambda}&=\frac{(\psi_{\lambda;i}-(q_{\lambda;i})'Y_{\lambda;i}-2q_{\lambda;i}(Y_{\lambda;i})_{z}}{\omega_{q_{\lambda}}}dz
%-2\omega_{q_{\lambda}}
%(Y_{\lambda;i})_{\overline{z}}d\overline{z}
%\end{align*}
%on $\mathcal{V}_{i,q_{\lambda}}$.
%\end{lemma}
One can easily check that $\Phi$ in Lemma \ref{lem:Douady-Hubbard}
is a well-defined closed one-form on $\tilde{M}_{q_{0}}$
which satisfies $(r_{q_{0}})^{*}(\Phi)=-\Phi$.

\begin{proof}
We calculate the $\lambda$-derivative at $\lambda=0$
with the notation in \S\ref{subsec:background-tangent-space}.
Notice that
\begin{align*}
\alpha_{i}(\lambda)\circ \alpha_{i}^{-1}(z_{i})
&=z_{i}-\lambda \tilde{X}_{i}(z_{i})-\overline{\lambda}\tilde{Y}_{i}(z_{i})+o(|\lambda|) \\
\tilde{q}_{\lambda;i}(z_{i})
&=\tilde{q}_{i}(z_{i})+\lambda \tilde{\phi}_{i}(z_{i})+
\overline{\lambda}\tilde{\psi}_{i}(z_{i})+o(|\lambda|)
\end{align*}
on $\mathbb{D}=\alpha_{i}(\mathcal{V}_{i,0})$
as $\lambda\to 0$,
where
$$
\tilde{Y}_{i}=-\left.\frac{\partial \alpha_{i}(\lambda)}{\partial \overline{\lambda}}\right|_{\lambda=0}\circ \alpha_{i}^{-1},\quad
\tilde{\psi}_{i}=\left.
\frac{\partial q_{\lambda;i}}{\partial \overline{\lambda}}
\right|_{\lambda=0},
$$
and the convergence is valid uniformly on any compact sets
of $\mathbb{D}$.
In our case,
$\lambda$-derivative of $D\ni\lambda\mapsto \douadyhubbard(q_{\lambda})(c)$
at $\lambda=0$
will be the desired formula.

Notice that the partial $\lambda$-
derivative of $\omega_{\lambda}$ at $\lambda=0$
is
$$
\left.
\frac{\partial \omega_{\lambda}}{\partial \lambda}
\right|_{\lambda=0}
=\frac{\tilde{\phi}_{i}}{2\omega_{q_{0}}}
$$
on $\mathbb{D}=\alpha_{i}(\mathcal{V}_{i,0})$.
Notice also in general that the derivative of
a function defined by the integration
$$
(-\delta,\delta)\ni s\mapsto \int_{a(s)}^{b(s)}f(s,x)dx
$$
at $s=0$
is equal to
$$
\int_{a(0)}^{b(0)}\frac{\partial f}{\partial s}(0,x)dx
+f(0,b(0))\frac{db}{ds}(0)-f(0,a(0))\frac{da}{ds}(0).
$$
Hence,
by the standard argument,
like as the discussion in Proposition 1 of \cite{MR0396936},
one can check the equations \eqref{eq:lem:Douady-Hubbard1}
hold.
\end{proof}

Though the following is well-known
(cf. \cite{MR866707},
\cite{MR1094714},
\cite{MR1135877}),
we confirm for the following for the completeness.
Indeed,
the calculation in the following proof will be a guide
in the later discussion.

\begin{lemma}[$\douadyhubbard$ defines a local coordinate]
\label{lem:douady-hubbard-local-coordinate}
Let $q_{0}\in \quaddiff{g,m}$ be as above.
When we take the above neighborhood $U$ of $q_{0}$
to be sufficiently small,
the mapping \eqref{eq:Douady-Hubbard} defines a holomorphic local coordinate around $q_{0}$.
\end{lemma}

\begin{proof}
%Rigorously speaking,
%we consider the 
%first ``equivariant''
%hypercohomology group $\mathbb{H}^{1}(L^{\bullet})$
%the complex of sheaves defined by the pull back sheaves
%$$
%\begin{CD}
%0 @>>> \mathcal{G}_{1}=r_{0}^{-1}(\Theta_{M_{q_{0}}}) @>{L_{\cdot}\omega_{q_{0}}^{2}}>> 
%\mathcal{G}_{2}=r_{0}^{-1}(\Omega_{M_{q_{0}}}^{\otimes 2})@>>>0
%\end{CD}
%$$
%instead of the hypercohomology group
%discussed in \S\ref{subsec:tangent-space-to-quadratic-differentials}.
%The sheaf
%$\mathcal{G}_{1}=r_{0}^{-1}(\Theta_{M_{q_{0}}})$
%is a subsheaf of the sheaf of germs of meromophic vector fields with (at most) first order
%poles at the preimages of zeros of $q_{0}$.
%The sheaf
%$\mathcal{G}_{2}=r_{0}^{-1}(\Omega_{M_{q_{0}}}^{\otimes 2})$
%is a subsheaf of the sheaf 
%of germs of holomorphic quadratic differentials
%which has (at least) second order zeros at the preimage of zeros of $q_{0}$.
%%associated
%%to the presheaf $\{\Gamma_{U}\}_{U}$,
%%where $U$ is an open set in $\tilde{M}_{q_{0}}$,
%%$$
%%\Gamma_{U}=\{\phi\in \Gamma(U,{\Omega_{\tilde{M}_{q_{0}}}}^{\otimes 2})\mid
%%(\phi)\ge (\omega_{q_{0}})\}
%%$$
%%and $(\omega_{q_{0}})$
%%is the divisor generated by $\omega_{q_{0}}$.
%For instance,
%for any $\phi\in \Gamma_{U}$,
%the quotient $\phi/\omega_{q_{0}}$ is a holomorphic $1$-form on $U$.
%
%Then,
%$\douadyhubbard(q_{\lambda})_{*}\left(\frac{\partial}{\partial \lambda}\right)$
%is the derivative of $\douadyhubbard$ on $U$
Let $v\in T_{q_{0}}\quaddiff{g,m}$ be a tangent vector
which is represented 
by the $1$ cocycle
$(\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})$
in 
$C^{0}(\mathcal{V}_{0},\Omega_{M_{q_{0}}}^{\otimes 2})\oplus
C^{1}(\mathcal{V}_{0},\Theta_{M_{q_{0}}})$,
where $\mathcal{V}_{0}=\{\{\pi_{q_{0}}(\mathcal{V}_{i,0})\}_{i}\}_{i}$
is a covering of $M_{q_{0}}$
(cf. \eqref{eq:subsec:background-tangent-space2}).
%and \eqref{eq:subsec:background-tangent-space3}).

Suppose that
$\douadyhubbard_{*}[v]=0$.
%$$
%\int_{c}\Phi_{0}=0,\quad \int_{c}\Psi_{0}=0
%$$
%for all $c\in H_{1}(\tilde{M}_{q_{0}})^{-}$
%from Lemma \ref{lem:Douady-Hubbard}.
%First we assume that
%$$
%\int_{c}\Phi_{0}=0\quad
%(c\in H_{1}(\tilde{M}_{q_{0}})^{-}).
%$$
Let $\Phi$ be a closed one form defined in Lemma \ref{lem:Douady-Hubbard}.
Since $(r_{0})^{*}(\Phi)=-\Phi$,
$$
\displaystyle \int_{c}\Phi=
\displaystyle \int_{r_{0}(c)}\Phi
=-\displaystyle \int_{c}\Phi
$$
for all $c\in H_{1}(\tilde{M}_{q_{0}})^{+}$,
where $H_{1}(\tilde{M}_{q_{0}})^{+}$ is the even part of the homology
of $\tilde{M}_{q}$ with coefficient in $\mathbb{Z}$.
Hence,
there is a smooth function $f$ on $\tilde{M}_{q_{0}}$ such that
$df=\Phi$
from de Rham's theorem.
Namely,
\begin{align*}
\frac{\tilde{\phi}_{i}}{2\omega_{q_{0}}}-\omega_{q_{0}}'\tilde{X}_{i}-
\omega_{q_{0}}\cdot (\tilde{X}_{i})_{z}
&=f_{z} \\
-\omega_{q_{0}}\cdot (\tilde{X}_{i})_{\overline{z}}
&=
f_{\overline{z}}
\end{align*}
hold on $\mathbb{D}=\alpha_{i}(\mathcal{V}_{i,0})$,
where $\tilde{\phi}_{i}$ and
$\tilde{X}_{i}$ are lifts of $\phi_{i}$ and $\tilde{X}_{i}$ to $\tilde{M}_{q_{0}}$,
respectively.
Since $\{(\mathcal{V}_{i,\lambda},\alpha_{i}(\lambda))\}_{i}$ is a
system of holomorphic local coordinates on $\tilde{M}_{q_{\lambda}}$
for each $\lambda\in D$,
$\dot{\nu}=-(\tilde{X}_{i})_{\overline{z}}$
defines a Beltrami differential
on $\tilde{M}_{q_{0}}$ which
descends to a Beltrami differential $\dot{\mu}$
on $M_{q_{0}}$.
Notice that
for any holomorphic quadratic differential $\varphi$ on $M_{q_{0}}$,
the quotient
$(\pi_{q_{0}})^{*}(\varphi)/\omega_{q_{0}}$ is a holomorphic $1$-form on $\tilde{M}_{q_{0}}$
and
\begin{align*}
\int_{M_{q_{0}}}\dot{\mu}\varphi
&=
\frac{1}{2}\int_{\tilde{M}_{q_{0}}}\dot{\nu}\cdot (\pi_{q_{0}})^{*}(\varphi)
=
\frac{1}{2}\int_{\tilde{M}_{q_{0}}}f_{\overline{z}}\frac{(\pi_{q_{0}})^{*}(\varphi)}{\omega_{q_{0}}}
\\
&=
\frac{1}{2}\int_{\tilde{M}_{q_{0}}}
\overline{\partial}
\left(f(z)\frac{(\pi_{q_{0}})^{*}(\varphi)(z)}{\omega_{q_{0}}(z)}dz
\right) \\
&=
\frac{1}{2}\int_{\tilde{M}_{q_{0}}}
d
\left(f(z)\frac{(\pi_{q_{0}})^{*}(\varphi)(z)}{\omega_{q_{0}}(z)}dz
\right)
=0.
\end{align*}
Hence,
$\dot{\mu}$ is an infinitesimally trivial Beltrami differenital
on $M_{q_{0}}$
by Teichm\"uller lemma (cf. Lemma 7.6 of \cite{MR1215481}).
This means that
the cohomology class
$$
[\{X_{ij}\}_{i,j}]\in 
H^{1}(\mathcal{V}_{0},\Theta_{M_{q_{0}}})
\cong H^{1}(M_{q_{0}},\Theta_{M_{q_{0}}})
$$
is trivial from \eqref{eq:subsec:background-tangent-space4}
since $\mathcal{V}_{0}$ is a Leray covering
(see also Theorem 3.5 of \cite{MR815922}).
Therefore,
there is a $0$-cochain $\{Z_{i}\}_{i}
\in C^{0}(\mathcal{V}_{0,i},\Theta_{M_{q_{0}}})$
such that $Z_{i}-Z_{j}=X_{ij}$
for all $i,j$.

Let $\zeta_{i}=L_{Z_{i}}(q_{0})
\in \Gamma(\mathcal{V}_{0,i},\Omega^{\otimes 2}_{M_{q_{0}}})$
and $\tilde{\zeta}_{i}=L_{\tilde{Z}_{i}}(\omega_{q_{0}}^{2})$
be the lift of $\zeta_{i}$.
Then,
$$
\tilde{\zeta}_{i}-\tilde{\zeta}_{j}=L_{\tilde{X}_{ij}}(\omega_{q_{0}}^{2})
$$
and hence $\{\tilde{\phi}_{i}-\tilde{\zeta}_{i}\}_{i}$ defines a holomorphic
quadratic differential $\tilde{\varphi}$ on $\tilde{M}_{q_{0}}$
%.
%Since $\{\phi_{i}-\zeta_{i}\}_{i}\in
%C^{0}(\tilde{M}_{q},r_{0}^{-1}\Omega^{\otimes 2}_{M_{q_{0}}})$,
%$\tilde{\varphi}=(\pi_{q_{0}})^{*}(\varphi)$
which descends to a holomorphic quadratic differential
$\varphi$ on $M_{q_{0}}$
associated to $\{\phi_{i}-\zeta_{i}\}_{i}$.
Since $\tilde{X}_{i}-\tilde{Z}_{i}=\tilde{X}_{j}-\tilde{Z}_{j}$ on $\mathcal{V}_{i,0}\cap \mathcal{V}_{j,0}$,
$\{\tilde{X}_{i}-\tilde{Z}_{i}\}_{i}$ defines a vector field $\tilde{W}$ on $\tilde{M}_{q_{0}}$
which satisfies
\begin{align*}
&((\omega_{q_{0}})'(\tilde{X}_{i})+
\omega_{q_{0}}\cdot (\tilde{X}_{i})_{z})dz
+
\omega_{q_{0}}\cdot (\tilde{X}_{i})_{\overline{z}}d\overline{z}
-
\frac{L_{\tilde{Z}_{i}}(\omega_{q_{0}}^{2})}{2\omega_{q_{0}}} \\
%&=
%\frac{(q_{i})'X_{i}+
%2q_{i}\cdot (X_{i})_{z}}{\omega_{q_{0}}}
%-
%\frac{(q_{i})'Z_{i}+
%2q_{i}\cdot (Z_{i})_{z}}{\omega_{q_{0}}} \\
%&=
%2(\omega_{q_{0}})'(X_{i}-Z_{i})+
%2\omega_{q_{0}}\cdot (X_{i}-Z_{i})_{z}dz
%+
%2\omega_{q_{0}}\cdot (X_{i}-Z_{i})_{\overline{z}}d\overline{z}
%\\
&=
\left((\omega_{q_{0}})'(\tilde{X}_{i}-\tilde{Z}_{i})+
\omega_{q_{0}}\cdot (\tilde{X}_{i}-\tilde{Z}_{i})_{z}
\right)dz
+
\omega_{q_{0}}\cdot (\tilde{X}_{i}-\tilde{Z}_{i})_{\overline{z}}d\overline{z} \\
&=d(\omega_{q_{0}}\cdot \tilde{W})
\end{align*}
since each $\tilde{Z}_{i}$ is a holomorphic section.
Consequently,
we deduce from the assumption that
\begin{equation}
\label{eq:lem:douady-hubbard-local-coordinate1}
0=
\int_{c}\Omega
=
\int_{c}\left(\Omega-\frac{\tilde{\zeta}_{i}-L_{\tilde{Z}_{i}}(\omega_{q_{0}}^{2})}{2\omega_{q_{0}}}
\right)
=
\int_{c}\frac{\tilde{\varphi}}{2\omega_{q_{0}}}
-
\int_{c}d(\omega_{q_{0}}\cdot W)
=
\int_{c}\frac{\tilde{\varphi}}{2\omega_{q_{0}}}
\end{equation}
for all $c\in H_{1}(\tilde{M}_{q_{0}})$,
and hence $\tilde{\varphi}=0$ and $\varphi=0$.
Therefore $\phi_{i}=\zeta_{i}$ and
$$
(\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})=
(\{L_{Z_{i}}(q_{0})\}_{i},\{Z_{i}-Z_{j}\}_{i,j}).
$$
Thus,
the cohomology class of the cochain $(\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})$
vanishes.
%By the same discussion,
%the condition $\int_{c}\Psi_{0}=0$ for all $c\in H_{1}(\tilde{M}_{q_{0}})^{-}$
%implies that
%the cohomology class of the cochain $(\{Y_{ij}\}_{i,j},\{\psi_{i}\}_{i})$
%vanishes.

As a consequence,
the derivative of the map $\douadyhubbard$ on
the tangent space
$T_{q_{0}}\quaddiff{g,m} \cong \mathbb{H}^{1}(L^{\bullet})$
is injective.
Since the dimensions of $\quaddiff{g,m}$ and ${\rm Hom}(H_{1}(\tilde{M}_{q_{0}})^{-},\mathbb{C})$ are same,
the derivative of $\douadyhubbard$ is an isomorphism
from $T_{q_{0}}\quaddiff{g,m}$
onto ${\rm Hom}(H_{1}(\tilde{M}_{q_{0}})^{-},\mathbb{C})$. 
\end{proof}

%From the proof of Lemma \ref{lem:douady-hubbard-local-coordinate},
%we obtain the following.
%
%\begin{corollary}[Douady-Hubbard]
%When the tangent vector represented 
%
%\end{corollary}


\subsection{Homomorphisms on slices}
\label{subsec:Hubbard-Masur-Slice}
For $F\in \mathcal{MF}-\{0\}$.
we define a slice
$$
\slice(F)=\{q\in \quaddiff{g,m}\mid \vertical(q)=F\}.
$$
for $F$ in $\quaddiff{g,m}$.
Hubbard and Masur showed that the restriction of the projection
$$
\slice(F)\ni q\mapsto \pi (q)\in \teich{g,m}
$$
is a homeomorphism
(\cite[\S2]{MR523212}).
The following two lemmas are well-known
(cf. \cite[Lemma 4.3]{MR523212} and \cite[Proposition 1]{MR1283559}.
See also \cite{2012arXiv1203.0273D}).
%(for instance,
%see Step 2 in the proof of Lemma 3.7 in \cite{MR523212}).
%However,
%we confirm them for the completeness.

\begin{lemma}
\label{lem:constant-representation}
Let $q_{0}\in \quaddiff{g,m}$ be a generic differential.
%and set $F_{0}=\vertical(q_{0})$.
When we take $U$ sufficiently small,
$$
\slice(\vertical(q_{0}))\cap U=\{q\in \quaddiff{g,m}\mid {\rm Re}(\douadyhubbard(q))=
{\rm Re}(\douadyhubbard(q_{0}))\}
$$
holds.
\end{lemma}
%
%\begin{proof}
%Since $q_{0}$ is generic,
%from Theorem 5.8 of \cite{2012arXiv1203.0273D},
%the map
%\begin{equation}
%\label{eq:Hubbard-Masur-map}
%U\ni q\mapsto \vertical(q)= |{\rm Re}(\sqrt{q})|\in \mathcal{MF}
%\end{equation}
%is a real analytic submersion around $q_{0}$
%(where $\mathcal{MF}$ admits a canonical PL structure
%inherited from the intersection number,
%and 
%$\vertical(q_{0})$ is a ``flat point'' with respect to
%the PL structure. See also \cite[Lemma 5.7]{2012arXiv1203.0273D} or the following discussion).
%Hence,
%when we take $U$ sufficiently small,
%there is a neighborhood $N_{1}$ of $\vertical(q_{0})$ in $\mathcal{MF}$
%such that 
%\eqref{eq:Hubbard-Masur-map}
%becomes a fiber space over $N_{1}$.
%By virtue of a result in \cite{Douady-Hubbard},
%the real part of the map \eqref{eq:Douady-Hubbard} 
%\begin{equation}
%\label{eq:Hubbard-Masur-map2}
%U\ni q\mapsto {\rm Re}(\douadyhubbard(q))\in 
%{\rm Hom}(H_{1}(\tilde{M}_{q_{0}})^{-},\mathbb{R})
%\end{equation}
%is also a submersion around $q_{0}$.
%Indeed,
%the differentials of the maps 
%\eqref{eq:Hubbard-Masur-map}
%and \eqref{eq:Hubbard-Masur-map2}
%at $q\in U$ give surjections from ${\rm Ker}(\pi_{*}\mid_{q})\cong \quaddiff{\pi(q)}$
%onto the tangent spaces of the targets.
%%and hence we recognize \eqref{eq:Hubbard-Masur-map2}
%%as a fiber space.
%
%By applying the discussion in \cite[Theorem 5.8]{2012arXiv1203.0273D} again,
%one can see that 
%the map \eqref{eq:Hubbard-Masur-map2}
%is constant on each fiber of the fibration \eqref{eq:Hubbard-Masur-map}.
%Indeed,
%take the
%train track $\tau_{q}$ and the triangulation $\Delta_{q}$
%on $M_{q}$ as \cite[Theorem 5.8]{2012arXiv1203.0273D}.
%Each edge of $\Delta_{q}$ is realized as a geodesic segment
%with respect to the flat structure defined by $q$.
%Let 
%$\tilde{\tau}_{q}$ and $\tilde{\Delta}_{q}$
%are lifts of $\tau_{q}$ and $\Delta_{q}$
%to $\tilde{M}_{q}$.
%We orient each edge of $\tilde{\Delta}_{q}$ such that
%${\rm Re}(\omega_{q})$ is  positive along the edge
%(we can choose $\Delta_{q}$ to have no vertical edge
%as \cite[Lemma 5.7]{2012arXiv1203.0273D}).
%By slightly opening
%lifts of each singular points of $q$ along vertical (short) segments
%directing the cusps of switches of $\tilde{\Delta}_{q}$,
%we get a tie neighborhood of $\tilde{\tau}_{q}$ which carries
%the vertical foliation $|{\rm Re}(\omega_{q})|$ of $\omega_{q}$.
%(cf. Figure \ref{fig:openning-cusp2}).
%\begin{figure}
%\includegraphics[height=3.5cm]{openning-cusp2.eps}
%\caption{Openning singular points to get a tie neighborhood.}
%\label{fig:openning-cusp2}
%\end{figure}
%Notice that each edge of $\Delta_{q}$
%is a tie of the tie neighborhood.
%When we take $U$ sufficiently small,
%we may assume that
%the natural marking $\tilde{M}_{q_{0}}\to \tilde{M}_{q}$
%induces identifications from
%$\tilde{\Delta}_{q_{0}}$ and $\tilde{\tau}_{q_{0}}$
%to
%$\tilde{\Delta}_{q}$ and $\tilde{\tau}_{q}$.
%
%Let $c\in H_{1}(\tilde{M}_{q_{0}})$.
%Suppose that $c$ can be realized as a union of
%polygonal loops
%with respect to the triangulation.
%Let $c=\sum_{k}a_{k}e_{k}$
%where $e_{k}$ is an (oriented) edge of $\tilde{\Delta}_{q_{0}}$
%and $a_{k}\in \mathbb{Z}$.
%Then,
%from the definition of the orientation of edges of the triangulation,
%one obtain
%$$
%{\rm Re}(\chi_{q}(c)
%)
%=
%\sum_{k}a_{k}
%{\rm Re}
%\int_{e_{k}}\omega_{q}
%=
%\sum_{k}a_{k}
%\int_{e_{k}}|{\rm Re}(\omega_{q})|.
%$$
%From the definition,
%the integration
%$\int_{e_{k}}|{\rm Re}(\omega_{q})|$
%is the intersection number of the tie $e_{k}$ of $\tilde{\tau}_{q}$
%with the measured foliation $|{\rm Re}(\omega_{q})|$,
%which is constant along each Hubbard-Masur slice in $U$
%since the measured foliation
%$|{\rm Re}(\omega_{q})|$ is the lift of the vertical foliation
%of $q$
%(See also \cite[\S4.3]{Mirzakhani}).
%In general,
%we have to apply the subdivision to the triangulation
%$\tilde{\Delta}_{q_{0}}$
%for realizing any $c\in H_{1}(\tilde{M}_{q_{0}})$.
%However,
%the same discussion is valid.
%Indeed,
%for an edge $e$ of the (midpoint) subdivition,
%the integration ${\rm Re}\int_{e}\omega_{q}$ is also constant
%along the fiber
%because as the above discussion
%the integrations of $|{\rm Re}(\omega_{q})|$
%over edges of $\tilde{\Delta}_{q_{0}}$ is constant,
%and the deformation of flat structures along the fiber is represented
%as a``vertical stretch'' on each triangle.
%
%This means that the maps \eqref{eq:Hubbard-Masur-map}
%and \eqref{eq:Hubbard-Masur-map2}
%determine the same fibration on $U$
%and hence we have done.
%%a neighborhood $N_{2}$ of $q_{0}$ in $\slice(\vertical(q_{0}))$
%%and a diffeomorphism
%%$$
%%\Xi\colon U\ni q\mapsto (\Xi_{1}(q),\Xi_{2}(q))\in N_{1}\times N_{2}
%%$$
%%such that $\Xi_{1}(q)=\vertical(q)$ for $q\in U$.
%%the diagram
%%$$
%%\begin{CD}
%%U\ni q @>{\Xi}>> (\vertical(q),\Xi_{2}(q))\in N_{1}\times N_{2}
%%\end{CD}
%%$$
%%
%%
%%
%%Suppose first that $F$ is essentially complete.
%%Following Lemma 4.3 in \cite{2012arXiv1203.0273D},
%%we construct a triangulation $\Delta_{q_{0}}$ of $M_{q_{0}}$ and
%%a bigon train track $\tau_{q_{0}}$ as follows
%%%\begin{enumerate}
%%%\item
%%%The vertices of $\Delta$ are singular points of $\phi$,
%%%the edges are saddle connections of $\phi$,
%%%and the triangulation $\Delta_{q_{0}}$
%%%is dual to the train track $\tau_{q_{0}}$,
%%%\item
%%%the foliation $\vertical(q_{0})$ is carried by $\tau_{q_{0}}$ in such a
%%%way each edge $e$ of $\Delta_{q_{0}}$ becomes a tie of the corresponding edge of
%%%$\tau_{q_{0}}$;
%%%in particular,
%%%\item
%%%the $|{\rm Re}\sqrt{q_{0}}|$-height of the edges of $\Delta_{q_{0}}$ gives the weight function on $\tau$ representing $\vertical(q_{0})$.
%%%\end{enumerate}
%%(See also Mirzakhani \cite[\S4.4]{Mirzakhani}).
%%%Indeed,
%%%notice that $\Delta_{q_{0}}$ has no vertical edge
%%%since $\vertical(q_{0})=F$ is essentially complete.
%%We set $\Delta_{q_{0}}$ as the Delaunay triangulation
%%with vertices the set of singular points of $q_{0}$.
%%The train track $\tau_{q_{0}}$ is obtained by modifying the dual graph to the triangulartion $\Delta_{q_{0}}$ at vertices of the dual graph.
%%Each triangle of $\Delta_{q_{0}}$
%%(a vertex of the dual graph)
%%has two horizontally short edges whose $|{\rm Re}\sqrt{q_{0}}|$-heights sum to that of the third edge.
%%We place a switch each each so that the incoming branches at the switch
%%are dual to the vertically short edges of the triangle.
%%If $\Delta_{q_{0}}$ has no horizontal edge,
%%the train track $\tau_{q_{0}}$ is a maximal bigon track.
%%In general,
%%
%%
%%
%%
%%The component of the complement of $\tau_{q_{0}}$
%%containing a zero is a trigon,
%%and that containing a pole is a bigon.
%%Hence, $\tau_{q_{0}}$ is a maximal bigon train track
%%(cf. \cite[\S3.4]{MR1144770}).
%%If we take $U$ sufficiently small if necessary,
%%we may assume that $\vertical(q)$ is carried by $\tau_{q_{0}}$
%%for all $q\in U$.
%%
%%Consider the lift $\tilde{\Delta}_{q_{0}}$ and $\tilde{\tau}_{q_{0}}$
%%of $\Delta_{q_{0}}$ and $\tau_{q_{0}}$
%%to $\tilde{M}_{q_{0}}$.
%%Then,
%%$|{\rm Re}(\omega_{q})|$ is carried by $\tilde{\tau}_{q_{0}}$
%%for all $q\in U$.
%%
%%
%\end{proof}

\begin{lemma}
\label{lem:real-analytic-section}
Let $x_{0}\in \teich{g,m}$ and $F\in \mathcal{MF}$.
If $q_{F,x_{0}}$ is generic,
then
the map
$$
\teich{g,m}\ni x\mapsto q_{F,x}\in \quaddiff{g,m}
$$
is real analytic around $x_{0}$.
\end{lemma}
%
%\begin{proof}
%%We take $U$ as Lemma \ref{lem:constant-representation}.
%%Then,
%This lemma follows from the observation that
%$$
%\quaddiff{g,m}\ni q\mapsto (\pi(q),\vertical(q))\in \teich{g,m}\times \mathcal{MF}
%$$
%is a real-analytic diffeomorphism onto its image
%around $q_{F,x_{0}}$
%as the discussion in Lemma \ref{lem:constant-representation}.
%\end{proof}



\section{Family of quadratic differentials with prescribed vertical foliation}
Fix $F\in \mathcal{MF}$.
Consider a family $\{q_{\lambda}\}_{\lambda\in D}$
of holomorphic quadratic differentials such that $\vertical(q_{\lambda})=F$.
Suppose that $q_{0}$ is generic
and $x_{\lambda}=\pi(q_{\lambda})\in \teich{g,m}$ varies holomorphically.
Then,
the family $\{q_{\lambda}\}_{\lambda\in D}$ is a real analytic family of quadratic
differentials
(cf. Lemma \ref{lem:real-analytic-section}).
Take a neighborhood $U$ of $q_{0}$ as the previous sections.
%Then,
%the vertical foliation
%$|{\rm Re}(\omega_{q_{\lambda}})|$ is constant on $D$.
%Therefore,
%\begin{align*}
%\frac{\partial \chi_{q_{\lambda}}(c)}{\partial \overline{\lambda}}
%&=
%i\frac{\partial}{\partial \overline{\lambda}}
%\left(
%{\rm Im}\int_{c}\omega_{q_{\lambda}}
%\right)
%=
%-2\frac{\partial}{\partial \overline{\lambda}}
%\left(
%\int_{c}\omega_{q_{\lambda}}
%-
%\overline{
%\int_{c}\omega_{q_{\lambda}}}
%\right)
%\end{align*}
%for all $c\in H_{1}(\tilde{M}_{q_{0}})$.

%
%\subsection{Hubbard-Masur slice}
%\label{subsec:Hubbard-Masur-Slice}
%For $F\in \mathcal{MF}-\{0\}$.
%%We call the subset
%we define
%$$
%\slice(F)=\{q\in \quaddiff{g,m}\mid \vertical(q)=F\}.
%$$
%%the \emph{Hubbard-Masur slice} of $F$ in $\quaddiff{g,m}$.
%Hubbard and Masur showed that the restriction of the projection
%$$
%\slice(F)\ni q\mapsto \pi (q)\in \teich{g,m}
%$$
%is a homeomorphism
%(\cite[\S2]{MR523212}).
%The following two lemmas might be well-known.
%However,
%we confirm them for the completeness.
%
%\begin{lemma}
%\label{lem:constant-representation}
%Let $q_{0}\in \quaddiff{g,m}$ be a generic differential.
%%and set $F_{0}=\vertical(q_{0})$.
%When we take $U$ sufficiently small,
%$$
%\slice(\vertical(q_{0}))\cap U=\{q\in \quaddiff{g,m}\mid {\rm Re}(\douadyhubbard(q))=
%{\rm Re}(\douadyhubbard(q_{0}))\}
%$$
%holds.
%\end{lemma}
%
%\begin{proof}
%Since $q_{0}$ is generic,
%from Theorem 5.8 of \cite{2012arXiv1203.0273D},
%the map
%\begin{equation}
%\label{eq:Hubbard-Masur-map}
%U\ni q\mapsto \vertical(q)= |{\rm Re}(\sqrt{q})|\in \mathcal{MF}
%\end{equation}
%is a real analytic submersion around $q_{0}$
%(where $\mathcal{MF}$ admits a canonical PL structure
%inherited from the intersection number,
%and 
%$\vertical(q_{0})$ is a ``flat point'' with respect to
%the PL structure. See also \cite[Lemma 5.7]{2012arXiv1203.0273D} or the following discussion).
%Hence,
%when we take $U$ sufficiently small,
%there is a neighborhood $N_{1}$ of $\vertical(q_{0})$ in $\mathcal{MF}$
%such that 
%\eqref{eq:Hubbard-Masur-map}
%becomes a fiber space over $N_{1}$.
%By virtue of a result in \cite{Douady-Hubbard},
%the real part of the map \eqref{eq:Douady-Hubbard} 
%\begin{equation}
%\label{eq:Hubbard-Masur-map2}
%U\ni q\mapsto {\rm Re}(\douadyhubbard(q))\in 
%{\rm Hom}(H_{1}(\tilde{M}_{q_{0}})^{-},\mathbb{R})
%\end{equation}
%is also a submersion around $q_{0}$.
%Indeed,
%the differentials of the maps 
%\eqref{eq:Hubbard-Masur-map}
%and \eqref{eq:Hubbard-Masur-map2}
%at $q\in U$ give surjections from ${\rm Ker}(\pi_{*}\mid_{q})\cong \quaddiff{\pi(q)}$
%onto the tangent spaces of the targets.
%%and hence we recognize \eqref{eq:Hubbard-Masur-map2}
%%as a fiber space.
%By applying the discussion in \cite[Theorem 5.8]{2012arXiv1203.0273D},
%one can see that 
%the map \eqref{eq:Hubbard-Masur-map2}
%is constant on each fiber of the fibration \eqref{eq:Hubbard-Masur-map}.
%This means that the maps \eqref{eq:Hubbard-Masur-map}
%and \eqref{eq:Hubbard-Masur-map2}
%determine the same fibration on $U$
%and hence we have done.
%%a neighborhood $N_{2}$ of $q_{0}$ in $\slice(\vertical(q_{0}))$
%%and a diffeomorphism
%%$$
%%\Xi\colon U\ni q\mapsto (\Xi_{1}(q),\Xi_{2}(q))\in N_{1}\times N_{2}
%%$$
%%such that $\Xi_{1}(q)=\vertical(q)$ for $q\in U$.
%%the diagram
%%$$
%%\begin{CD}
%%U\ni q @>{\Xi}>> (\vertical(q),\Xi_{2}(q))\in N_{1}\times N_{2}
%%\end{CD}
%%$$
%%
%%
%%
%%Suppose first that $F$ is essentially complete.
%%Following Lemma 4.3 in \cite{2012arXiv1203.0273D},
%%we construct a triangulation $\Delta_{q_{0}}$ of $M_{q_{0}}$ and
%%a bigon train track $\tau_{q_{0}}$ as follows
%%%\begin{enumerate}
%%%\item
%%%The vertices of $\Delta$ are singular points of $\phi$,
%%%the edges are saddle connections of $\phi$,
%%%and the triangulation $\Delta_{q_{0}}$
%%%is dual to the train track $\tau_{q_{0}}$,
%%%\item
%%%the foliation $\vertical(q_{0})$ is carried by $\tau_{q_{0}}$ in such a
%%%way each edge $e$ of $\Delta_{q_{0}}$ becomes a tie of the corresponding edge of
%%%$\tau_{q_{0}}$;
%%%in particular,
%%%\item
%%%the $|{\rm Re}\sqrt{q_{0}}|$-height of the edges of $\Delta_{q_{0}}$ gives the weight function on $\tau$ representing $\vertical(q_{0})$.
%%%\end{enumerate}
%%(See also Mirzakhani \cite[\S4.4]{Mirzakhani}).
%%%Indeed,
%%%notice that $\Delta_{q_{0}}$ has no vertical edge
%%%since $\vertical(q_{0})=F$ is essentially complete.
%%We set $\Delta_{q_{0}}$ as the Delaunay triangulation
%%with vertices the set of singular points of $q_{0}$.
%%The train track $\tau_{q_{0}}$ is obtained by modifying the dual graph to the triangulartion $\Delta_{q_{0}}$ at vertices of the dual graph.
%%Each triangle of $\Delta_{q_{0}}$
%%(a vertex of the dual graph)
%%has two horizontally short edges whose $|{\rm Re}\sqrt{q_{0}}|$-heights sum to that of the third edge.
%%We place a switch each each so that the incoming branches at the switch
%%are dual to the vertically short edges of the triangle.
%%If $\Delta_{q_{0}}$ has no horizontal edge,
%%the train track $\tau_{q_{0}}$ is a maximal bigon track.
%%In general,
%%
%%
%%
%%
%%The component of the complement of $\tau_{q_{0}}$
%%containing a zero is a trigon,
%%and that containing a pole is a bigon.
%%Hence, $\tau_{q_{0}}$ is a maximal bigon train track
%%(cf. \cite[\S3.4]{MR1144770}).
%%If we take $U$ sufficiently small if necessary,
%%we may assume that $\vertical(q)$ is carried by $\tau_{q_{0}}$
%%for all $q\in U$.
%%
%%Consider the lift $\tilde{\Delta}_{q_{0}}$ and $\tilde{\tau}_{q_{0}}$
%%of $\Delta_{q_{0}}$ and $\tau_{q_{0}}$
%%to $\tilde{M}_{q_{0}}$.
%%Then,
%%$|{\rm Re}(\omega_{q})|$ is carried by $\tilde{\tau}_{q_{0}}$
%%for all $q\in U$.
%%
%%
%\end{proof}
%
%\begin{lemma}
%\label{lem:real-analytic-section}
%Let $x_{0}\in \teich{g,m}$ and $F\in \mathcal{MF}$.
%If $q_{F,x_{0}}$ is generic,
%then
%the map
%$$
%\teich{g,m}\ni x\mapsto q_{F,x_{0}}\in \quaddiff{g,m}
%$$
%is real analytic around $x_{0}$.
%\end{lemma}
%
%\begin{proof}
%%We take $U$ as Lemma \ref{lem:constant-representation}.
%%Then,
%This lemma follows from the observation that
%$$
%\quaddiff{g,m}\ni q\mapsto (\pi(q),\vertical(q))\in \teich{g,m}\times \mathcal{MF}
%$$
%is a real-analytic diffeomorphism onto its image
%around $q_{F,x_{0}}$
%as the discussion in Lemma \ref{lem:constant-representation}.
%\end{proof}

\subsection{Differential via the complex conjugate}
\label{subsec:Differentical-via-the-complex-conjugate}
%Let $F\colon \mathbb{D}\to M$ be a $C^{1}$-mapping to a complex manifold
%$M$.
%
We first notice a simple observation.
Let $\pi\colon E\to M$ be a holomorphic vector bundle.
%between complex manifolds $M_{1}$ and $M_{2}$,
Let $g\colon \mathbb{D}\to M$ be a holomorphic mapping
and $G\colon \mathbb{D}\to E$ be a $C^{1}$ mapping
such that $\pi\circ G=g$ on $\mathbb{D}$.
Then,
$$
\pi_{*}\left(G_{*}
\left(\frac{\partial}{\partial \overline{\lambda}}\right)
\right)
=g_{*}
\left(\frac{\partial}{\partial \overline{\lambda}}\right)
=0
$$
since $g$ is holomorphic.
Hence,
\begin{equation}
\label{eq:fiber-partial}
\left.G_{*}\left(\frac{\partial}{\partial \overline{\lambda}}
\right)
\right|_{\lambda}\in {\rm Ker}(\pi_{*})\cong E_{g(\lambda)}
\end{equation}
for all $\lambda\in \mathbb{D}$.
%
%
%
%
%\subsection{}
%Henceforth,
%we shall continue to use the symbols $q_{0}$, $U$,...
%which are defined and taken in \S\ref{sec:local-coordinate}.
\begin{lemma}[First derivatives]
\label{lem:transversal}
Let $\{q_{\lambda}\}_{\lambda\in D}$ be 
the family of holomorphic quadratic differentials defined as above.
%a $C^{k+1}$-family
%of holomorphic quadratic differentials passing a generic differential at $\lambda=0$
%such that $q_{\lambda}\in U$ for all $\lambda$
%Suppose that $\mathbb{D}\ni \lambda\mapsto \pi(q_{\lambda})\in \teich{g,m}$ is holomorphic.
Let $\eta_{\lambda}$ be a holomorphic quadratic differential
on $X_{q_{\lambda}}$
such that the infinitesimal Bertrami differential
$\dot{\mu}$
associated to the tangent vector 
for the $\lambda$-derivative of the mapping
$$
D\ni \lambda\mapsto \pi(q_{\lambda})\in \teich{g,m}
$$
at $\lambda$ satisfies
$$
\int_{M_{q_{\lambda}}}\dot{\mu}\psi
=
\int_{M_{q_{\lambda}}}\frac{\overline{\eta_{\lambda}}}{|q_{\lambda}|}
\psi
$$
for all holomorphic quadratic differential $\psi$ on $M_{q_{\lambda}}$.
Then,
%$\lambdais a $C^{k}$-family of quadratic differentials
%$\{\eta_{\lambda}\}_{\lambda\in D}$
%such that $\eta_{\lambda}\in \quaddiff{\pi(q_{\lambda})}$
%and
$$
\frac{\partial\,\chi_{q_{\lambda}}(c)}{\partial \lambda}
=\int_{c}
\overline{
\left(
\frac{(\pi_{q_{\lambda}})^{*}(\eta_{\lambda})}{\omega_{q_{\lambda}}}
\right)
},
\quad
\frac{\partial\,\chi_{q_{\lambda}}(c)}{\partial \overline{\lambda}}
=-\int_{c}\frac{(\pi_{q_{\lambda}})^{*}(\eta_{\lambda})}{\omega_{q_{\lambda}}}
$$
for any $c\in H_{1}(\tilde{M}_{q_{0}})$ and $\lambda\in D$.
\end{lemma}

\begin{proof}
The existence and the uniqueness of $\eta_{\lambda}$ follows from
an argumant by David Dumas \cite[Theorem 5.3]{2012arXiv1203.0273D}
since each $q_{\lambda}$ is generic.
Indeed,
the pairing
$$
(\eta,\psi)\mapsto \int_{M_{q_{\lambda}}}\frac{\overline{\eta}\psi}{|q_{\lambda}|}
$$
is a positive definite hermitian inner product
on the space of holomorphic quadratic differentials on $M_{q_{\lambda}}$.
Notice that
$$
\frac{\partial\,\chi_{q_{\lambda}}(c)}{\partial \overline{\lambda}}
=
0=\int_{c}\frac{(\pi_{q_{\lambda}})^{*}(\eta_{\lambda})}{\omega_{q_{\lambda}}}
$$
for any $c\in H_{1}(\tilde{M}_{q_{0}})^{+}$ and $\lambda\in D$.
It suffices to show the case where $\lambda=0$
and $c\in H_{1}(\tilde{M}_{q_{0}})^{-}$.
As we discussed around \eqref{eq:fiber-partial},
from Lemma \ref{lem:Douady-Hubbard}
and the proof of Lemma \ref{lem:douady-hubbard-local-coordinate},
there is a holomorphic quadratic differential $\varphi$ on $X_{q_{0}}$
such that
$$
\left.
\frac{\partial\,\chi_{q_{\lambda}}(c)}{\partial \overline{\lambda}}
\right|_{\lambda=0}
=\int_{c}\frac{(\pi_{q_{0}})^{*}(\varphi)}{\omega_{q_{0}}}
$$
for any $c\in H_{1}(\tilde{M}_{q_{0}})^{-}$
(see \eqref{eq:lem:douady-hubbard-local-coordinate1}).
Since the vertical foliation $|{\rm Re}(\omega_{q_{\lambda}})|$
of $\omega_{q_{\lambda}}$ is unchanged
on $D$,
for all $c\in H_{1}(\tilde{M}_{q_{0}})^{-}$,
the real part of $\chi_{q_{\lambda}}(c)$ is a constant function on $D$
by Lemma \ref{lem:constant-representation}.
Therefore,
$$
\frac{\partial\,\chi_{q_{\lambda}}(c)}{\partial \overline{\lambda}}
=-
\overline{\frac{\partial\,\chi_{q_{\lambda}}(c)}{\partial \lambda}}
$$
on $D$.
%Fix an appropriate covering $\mathcal{U}$ of $M_{q_{0}}$.
Let $(\{\phi_{i}\}_{i},\{X_{ij}\}_{i,j})$
be the cocycle representing the tangent vector
associated to the $\lambda$-derivative
of the mapping $D\ni \lambda\mapsto q_{\lambda}$
at $\lambda=0$.
Let $\{X_{i}\}_{i}$ be a $0$-cochain of the sheaf
of $C^{\infty}$-vector fields with $X_{i}-X_{j}=X_{ij}$.
Then,
$$
\Omega=\left(\frac{\tilde{\phi}_{i}}{2\omega_{q_{0}}}
-\omega_{q_{0}}'\tilde{X}_{i}-\omega_{q_{0}}(\tilde{X}_{i})_{z}
\right)dz
-
(\omega_{q_{0}}(\tilde{X}_{i})_{\overline{z}})
d\overline{z}
$$
on $U_{i}$ defines a closed $1$-form on $\tilde{M}_{q_{0}}$
satisfies
$$
\frac{\partial\,\chi_{q_{\lambda}}(c)}{\partial \lambda}
=\int_{c}\Omega
$$
for all $c\in H_{1}(\tilde{M}_{q_{0}})$.
Therefore,
there is a $C^{\infty}$-function $f$ on $\tilde{M}_{q_{0}}$
such that
$$
\Omega=-\overline{\frac{(\pi_{q_{0}})^{*}(\varphi)(z)}{\omega_{q_{0}}(z)}}d\overline{z}+df,
$$
that is,
\begin{align}
\frac{\tilde{\phi}_{i}}{2\omega_{q_{0}}}
-\omega_{q_{0}}'\tilde{X}_{i}-\omega_{q_{0}}(\tilde{X}_{i})_{z}
&=f_{z}
\label{eq:lem:transversal1}
\\
-\omega_{q_{0}}(\tilde{X}_{i})_{\overline{z}}
&=
-\overline{\left(\frac{(\pi_{q_{0}})^{*}(\varphi)(z)}{\omega_{q_{0}}(z)}\right)}
+f_{\overline{z}}
\label{eq:lem:transversal2}
\end{align}
on $\mathcal{V}_{i,0}$.
From
%the definition of $\eta_{0}$
%and 
\eqref{eq:lem:transversal2},
the lift $(\pi_{q_{0}})^{*}(\dot{\mu})$ of
$\dot{\mu}=-(X_{i})_{\overline{z}}$
satisfies
\begin{equation}
\label{eq:lem:transversal3}
(\pi_{q_{0}})^{*}(\dot{\mu})
%\frac{\overline{(\pi_{q_{0}})^{*}(\eta_{0})}}{|\omega_{q_{0}}|^{2}}
=
-\frac{\overline{(\pi_{q_{0}})^{*}(\varphi)}}{|\omega_{q_{0}}|^{2}}
+\frac{f_{\overline{z}}}{\omega_{q_{0}}}
\end{equation}
on $\tilde{M}_{q_{0}}$.
Let $\psi$ be a holomorphic quadratic differenital on $M_{q_{0}}$.
From \eqref{eq:lem:transversal3}
\begin{align*}
\int_{\tilde{M}_{q_{0}}}
%\frac{\overline{(\pi_{q_{0}})^{*}(\eta_{0})}}{|\omega_{q_{0}}|^{2}}
(\pi_{q_{0}})^{*}(\dot{\mu})
(\pi_{q_{0}})^{*}(\psi)
&=
-\int_{\tilde{M}_{q_{0}}}
\frac{\overline{(\pi_{q_{0}})^{*}(\varphi)}}{|\omega_{q_{0}}|^{2}}
(\pi_{q_{0}})^{*}(\psi)
+
\int_{\tilde{M}_{q_{0}}}
\frac{f_{\overline{z}}}{\omega_{q_{0}}}
(\pi_{q_{0}})^{*}(\psi) \\
&=
-\int_{\tilde{M}_{q_{0}}}
\frac{\overline{(\pi_{q_{0}})^{*}(\varphi)}}{|\omega_{q_{0}}|^{2}}
(\pi_{q_{0}})^{*}(\psi)
+
\int_{\tilde{M}_{q_{0}}}
d\left(
f\frac{(\pi_{q_{0}})^{*}(\psi)}{\omega_{q_{0}}}
\right) \\
&=
-\int_{\tilde{M}_{q_{0}}}
\frac{\overline{(\pi_{q_{0}})^{*}(\varphi)}}{|\omega_{q_{0}}|^{2}}
(\pi_{q_{0}})^{*}(\psi)
\end{align*}
By descending to $M_{q_{0}}$,
we have
$$
\int_{M_{q_{0}}}\frac{\overline{\eta_{0}}}{|q_{0}|}
\psi
=
\int_{M_{q_{0}}}\dot{\mu}\psi
=
-\int_{M_{q_{0}}}\frac{\overline{\varphi}\psi}{|q_{0}|}
$$
for all $\psi$.
Therefore,
$\varphi=-\eta_{0}$ from 
\cite[Theorem 5.3]{2012arXiv1203.0273D}
again.
%since $q_{0}$ is generic.
\end{proof}

\subsection{Laplacian of homomorphisms}
This section is devoted to calculating
the Laplacian of Douady-Hubbard map
\eqref{eq:Douady-Hubbard}.
The author must confess that
for the proof of the plurisubharmonicity of extremal length funcitons,
we need the equation \eqref{eq:Omega-0-Laplacian}
rather than the formula \eqref{lem:transversal2-1} in the following lemma.
However,
we give the following lemma for its own interests.

\begin{lemma}[Laplacian]
\label{lem:transversal2}
Under the assumption in Lemma \ref{lem:transversal2},
%For $v\in T_{x_{0}}\teich{g,m}$,
%we define $\eta_{v}\in \quaddiff{x_{0}}$ as
%$$
%\langle v,\varphi\rangle=\int_{M_{q_{0}}}\frac{\overline{\eta_{v}}}{|q_{0}|}\varphi
%$$
%holds for all $\varphi\in \quaddiff{x_{0}}$.
we have
\begin{equation}
\label{lem:transversal2-1}
\left.
\frac{\partial^{2}(\chi_{q_{\lambda}}(c))}{\partial \lambda\partial\overline{\lambda}}
\right|_{\lambda=0}
=
-4i\,{\rm Im}
\int_{\tilde{M}_{q_{0}}}
\frac{|(\pi_{q_{0}})^{*}(\eta_{0})|^{2}}{|\omega_{q_{0}}|^{2}}
\frac{\sigma_{c}}{\omega_{q_{0}}}
\end{equation}
for $c\in H_{1}(\tilde{M}_{q_{0}})$,
where $\sigma_{c}=\sigma_{c}(z)dz$ is the holomorphic part of the
reproducing harmonic differential
associated to $c$.
\end{lemma}

\begin{proof}
When the homology class $c$ is in the even homology group,
$\chi_{q_{\lambda}}(c)=0$.
The integrand of the right-hand side of \eqref{lem:transversal2-1}
satisfies
$$
r_{0}^{*}\left(\frac{|(\pi_{q_{0}})^{*}(\eta_{0})|^{2}}{|\omega_{q_{0}}|^{2}}
\frac{\sigma_{c}}{\omega_{q_{0}}}\right)
=
\frac{|(\pi_{q_{0}})^{*}(\eta_{0})|^{2}}{|-\omega_{q_{0}}|^{2}}
\frac{\sigma_{c}}{-\omega_{q_{0}}}
=
-
\frac{|(\pi_{q_{0}})^{*}(\eta_{0})|^{2}}{|\omega_{q_{0}}|^{2}}
\frac{\sigma_{c}}{\omega_{q_{0}}}
$$
and hence,
the integral over $\tilde{M}_{q_{0}}$ vanishes.
This means that \eqref{lem:transversal2-1} holds for homology classes
in the even homology group.

By applying the same argument as that of Lemma \ref{lem:Douady-Hubbard},
one obtain
$$
%\left.
%\triangle (\chi_{q_{\lambda}}(c))
%\right|_{\lambda=0}
%=
\left.
\frac{\partial^{2}\,\chi_{q_{\lambda}}(c)}{\partial\lambda\partial \overline{\lambda}}
\right|_{\lambda=0}
=\int_{c}\Omega_{0}
$$
for any $c\in H_{1}(\tilde{M}_{q_{0}})$,
where $\Omega_{0}$ is a closed differential of the form
\begin{align*}
\left(
\dot{\eta}_{0}
+\left(\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)'(\tilde{X}_{i})
+
\left(\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)(\tilde{X}_{i})_{z}
\right)dz 
%\\
%&\quad
+
\left(
\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)(\tilde{X}_{i})_{\overline{z}}
d\overline{z}
\end{align*}
and
$$
\dot{\eta}_{0}
=
-\left.
\frac{\partial}{\partial\lambda}\left(\frac{(\pi_{q_{\lambda}})^{*}(\eta_{\lambda})}{\omega_{q_{\lambda}}}\right)
\right|_{\lambda=0}.
$$
Since the real part of
the function $D\ni \lambda\mapsto \chi_{q_{\lambda}}(c)$
is constant
and the Laplacian is the real operator,
$$
\int_{c}\Omega_{0}=
\left.
\frac{\partial^{2}\,\chi_{q_{\lambda}}(c)}{\partial\lambda\partial \overline{\lambda}}
\right|_{\lambda=0}
=
-\overline{\left.
\frac{\partial^{2}\,\chi_{q_{\lambda}}(c)}{\partial\lambda\partial \overline{\lambda}}
\right|_{\lambda=0}
}
=
-\overline{\left(\int_{c}\Omega_{0}\right)}
=-\int_{c}\overline{\Omega_{0}}
$$
for all $c\in H_{1}(\tilde{M}_{q_{0}})$.
Hence,
there is a $C^{\infty}$-function $f$ on $\tilde{M}_{q_{0}}$
such that
$$
\Omega_{0}=-\overline{\Omega_{0}}+df.
$$
Namely,
we have
%\begin{align*}
%&
\begin{align*}
\dot{\eta}_{0}
+\left(\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)'(X_{i})
+
\left(\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)(\tilde{X}_{i})_{z}
&=
-\overline{
\left(
\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)(\tilde{X}_{i})_{\overline{z}}}
+f_{z} \\
&=
\overline{
\left(
\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)
}
(\pi_{q_{0}})^{*}(\dot{\mu})
+f_{z},
\end{align*}
%,\quad
%\mbox{and}
%\\
%and
%$$
%\overline{
%\dot{\eta}_{0}
%-\left(\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)'(X_{i})
%-
%\left(\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)(\tilde{X}_{i})_{z}
%}
%=
%\left(
%\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)
%(\tilde{X}_{i})_{\overline{z}}
%+f_{\overline{z}}.
%$$
%\begin{equation}
%(\tilde{X}_{i})_{\overline{z}}
%=\frac{\overline{(\pi_{q_{0}})^{*}(\eta_{0})}}{2|\omega_{q_{0}}|^{2}}
%+\frac{h_{\overline{z}}}{2\omega_{q_{0}}}.
%\end{equation}
%From the above two equations,
%we have $\overline{f_{z}}=f_{\overline{z}}$,
%or $(f-\overline{f})_{\overline{z}}=0$.
%Hence,
%the imaginary part of $f$ is holomorphic,
%which means that the imaginary part of $f$ is constant.
%Thus,
%we may assume that $f$ is a real valued function.
where $\dot{\mu}=-(X_{i})_{\overline{z}}$ is the infinitesimal Beltrami
differential representing the tangent vector 
of the mapping $D\ni \mapsto \pi(q_{\lambda})$
at $\lambda=0$
(cf. \eqref{eq:infinitesimal1}).
Applying the above equation,
we also obtain
\begin{align}
\Omega_{0}
&=
\left(
\overline{
\left(
\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)
(\pi_{q_{0}})^{*}(\dot{\mu})
}
+f_{z}
\right)dz
-
\left(
\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)
(\pi_{q_{0}})^{*}(\dot{\mu})
d\overline{z}
\nonumber
\\
&=
%\left(
\overline{
\left(
\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)
(\pi_{q_{0}})^{*}(\dot{\mu})
}
dz
-
\left(
\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)
(\pi_{q_{0}})^{*}(\dot{\mu})
d\overline{z}
%\right)
\label{eq:Omega-0-Laplacian}
\\
&\qquad\qquad+\frac{1}{2}df+\frac{i}{2}\,{}^{*}df.
\nonumber
%&=
%\left(
%\frac{|(\pi_{q_{0}})^{*}(\eta_{0})|^{2}}{\overline{\omega_{q_{0}}}|\omega_{q_{0}}|^{2}}
%+
%\left(
%\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\right)
%\frac{h_{\overline{z}}}{\omega_{q_{0}}}
%+f_{z}
%\right)dz
%-
%\frac{|(\pi_{q_{0}})^{*}(\eta_{0})|^{2}}{\omega_{q_{0}}|\omega_{q_{0}}|^{2}}
%d\overline{z} \\
%&=
%-4i\,
%{\rm Im}
%%\frac{|(\pi_{q_{0}})^{*}(\eta_{0})|^{2}}{|\omega_{q_{0}}|^{4}}
%%{\rm Im}(\omega_{q_{0}})
%\left(
%\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}
%(\pi_{q_{0}})^{*}(\dot{\mu})\,
%d\overline{z}
%\right)
%+f_{z}dz
%\\
%&=
%-4i\,
%{\rm Im}
%%\frac{|(\pi_{q_{0}})^{*}(\eta_{0})|^{2}}{|\omega_{q_{0}}|^{4}}
%%{\rm Im}(\omega_{q_{0}})
%\left(
%\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}
%(\pi_{q_{0}})^{*}(\dot{\mu})\,
%d\overline{z}
%\right)
%+\frac{1}{2}df+\frac{i}{2}\,{}^{*}df
\end{align}
Let $\sigma(c)=\sigma_{c}(z)dz+\overline{\sigma_{c}(z)}d\overline{z}$
be the reproducing harmonic differential on $\tilde{M}_{q_{0}}$
associated to $c\in H_{1}(\tilde{M}_{q_{0}})$,
where $\sigma_{c}(z)dz$ is a holomorphic $1$-form on $\tilde{M}_{q_{0}}$
(we call this holomorphic $1$-form the \emph{holomorphic part} of $\sigma(c)$).
Namely,
the differential $\sigma(c)$ is a unique harmonic differential on $\tilde{M}_{q_{0}}$
which satisfies
$$
\int_{c}\omega=\int_{\tilde{M}_{q_{0}}}\omega\wedge {}^{*}\overline{\sigma(c)}
$$
for all closed $1$-form $\omega$ on $\tilde{M}_{q_{0}}$.
Then,
by applying
the orthogonal decomposition theorem
for the space of $L^{2}$-closed forms,
we have
\begin{align*}
\left.
\frac{\partial^{2}\,\chi_{q_{\lambda}}(c)}{\partial\lambda\partial \overline{\lambda}}
\right|_{\lambda=0}
%&=-32i\int_{\tilde{M}_{q_{0}}}
%{\rm Im}\left(
%\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}
%(\pi_{q_{0}})^{*}(\dot{\mu})\,
%d\overline{z}
%\right)
%\wedge
%{}^{*}\overline{\sigma_{c}} \\
&=
\int_{c}\Omega_{0}
=\int_{\tilde{M}_{q_{0}}}\Omega_{0}\wedge {}^{*}\overline{\sigma(c)}\\
&=\int_{\tilde{M}_{q_{0}}}
\left(
\overline{\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}
(\pi_{q_{0}})^{*}(\dot{\mu})}\,
dz-
\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}
(\pi_{q_{0}})^{*}(\dot{\mu})\,
d\overline{z}
\right) \\
&\quad\quad\qquad\qquad
\wedge
{}^{*}\overline{(\sigma_{c}(z)dz+\overline{\sigma_{c}(z)}d\overline{z})}
\\
&=
\int_{\tilde{M}_{q_{0}}}
\left(
\overline{\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}
(\pi_{q_{0}})^{*}(\dot{\mu})}\,
dz-
\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}
(\pi_{q_{0}})^{*}(\dot{\mu})\,
d\overline{z}
\right) \\
&\quad\quad\qquad\qquad
\wedge
(-i\overline{\sigma_{c}}d\overline{z}+i\sigma_{c}dz)
\\
&=
-4i\,{\rm Im}
\int_{\tilde{M}_{q_{0}}}
\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}
(\pi_{q_{0}})^{*}(\dot{\mu})\sigma_{c}.
\end{align*}
Now,
we suppose that the homology class $c$ is in $H_{1}(\tilde{M}_{q_{0}})^{-}$.
Since $(r_{0})_{*}(c)=-c$,
$r_{0}^{*}\sigma(c)=-\sigma(c)$.
This means that $\sigma(c)$ has zeroes at each branch points of
$\pi_{q_{0}}\colon
\tilde{M}_{q_{0}}\to M_{q_{0}}$
and the holomorphic quadratic differential
$$
\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}\sigma_{c}=
\frac{(\pi_{q_{0}})^{*}(\eta_{0})(z)}{\omega_{q_{0}}(z)}\sigma_{c}(z)dz^{2}
$$
descends to a holomorphic quadratic differential on $M_{q_{0}}$.
Therefore,
we obtain
from the definition of $\eta_{0}$
that
\begin{align}
\left.
\frac{\partial^{2}\,\chi_{q_{\lambda}}(c)}{\partial\lambda\partial \overline{\lambda}}
\right|_{\lambda=0}
&=
-4i\,{\rm Im}
\int_{\tilde{M}_{q_{0}}}
\frac{(\pi_{q_{0}})^{*}(\eta_{0})}{\omega_{q_{0}}}
\frac{\overline{(\pi_{q_{0}})^{*}(\eta_{0})}}{|\omega_{q_{0}}|^{2}}\sigma_{c}
\nonumber
\\
&=
-4i\,{\rm Im}
\int_{\tilde{M}_{q_{0}}}
\frac{|(\pi_{q_{0}})^{*}(\eta_{0})|^{2}}{|\omega_{q_{0}}|^{2}}
\frac{\sigma_{c}}{\omega_{q_{0}}}
\label{eq:laplacian-1}
\end{align}
(cf. Lemma \ref{lem:transversal}).
This is what we wanted.
\end{proof}

%{\color{red}
%Period-relationは
%$$
%\|\theta\|^{2}=i\sum_{i=1}^{g}\left[
%\int_{a_{i}}\theta\overline{\int_{b_{i}}\theta}
%-
%\int_{b_{i}}\theta\overline{\int_{a_{i}}\theta}
%\right]
%$$
%である．
%}

\section{Levi forms and Plurisubharmonicity of subspecies}
\subsection{Levi forms of extremal length functions}
For a real-valured $C^{2}$-function  $U$  on a complex manifold $N$,
the \emph{Levi form} of $U$ is an hermitian inner product on
the holomorphic tangent bundle $TN$ of $N$ defined as
$$
\levi{U}[v,\overline{v}]
%\partial \overline{\partial}U
=\sum_{i,j=1}^{n}
\frac{\partial^{2}U}{\partial z_{i}\partial \overline{z}_{j}}
v_{i}\overline{v_{j}}
$$
for $v=\sum_{i=1}^{n}v_{i}(\partial/\partial z_{i})\in TN$.
Let $p\in N$ and $v\in T_{p}N$.
Let $F\colon \mathbb{D}\to N$ be a holomorphic mapping
with $f(0)=p$ and $f_{*}(\partial/\partial\lambda\mid_{\lambda=0})=v$,
then
$$
%\partial \overline{\partial}U[v,\overline{v}]
\levi{U}[v,\overline{v}]=
\left.
\frac{\partial^{2}(U\circ f)}{\partial\lambda\partial\overline{\lambda}}
\right|_{\lambda=0}.
$$
\begin{theorem}[Levi forms]
\label{thm:leviform}
Let $x_{0}=(M_{0},f_{0})\in \teich{g,m}$ and $v=[\dot{\mu}]\in T_{x_{0}}\teich{g,m}$.
Let $F\in \mathcal{MF}$.
Suppose that the Hubbard-Masur differential $q_{0}=q_{F,x_{0}}$ is generic.
Then,
the extremal length function is real analytic around $x_{0}$,
and
the Levi form of the extremal length function
$\teich{g,m}\ni x\mapsto \ext_{x}(F)$
at $x_{0}$ satisfies
$$
\levi{\ext_{\cdot }(F)}[v,\overline{v}]
=
2\int_{M_{0}}\frac{|\eta_{v}|^{2}}{|q_{0}|},
$$
where $\eta_{v}$ is a unique
holomorphic quadratic differential on $M_{0}=M_{q_{0}}$
satisfying that
$$
\langle v,\psi\rangle=
\int_{M_{0}}\dot{\mu}\psi=
\int_{M_{0}}\frac{\overline{\eta_{v}}}{|q_{0}|}\psi
$$
for all $\psi\in \quaddiff{x_{0}}$.
\end{theorem}

\begin{remark}
In \S\ref{sec:geometric-interpretation},
we will discuss a geometric interpretation
of the anti-complex linear map
$$
T_{x_{0}}\teich{g,m}\ni
v\mapsto \eta_{v}
\in \quaddiff{x_{0}}.
$$
\end{remark}


\begin{proof}[Proof of Theorem \ref{thm:leviform}]
%For the simplicity,
%let $A_{c}=(\chi_{q_{\lambda}}(c))_{\lambda}|_{\lambda=0}$.
%Since the real part of $\chi_{q_{\lambda}}(c)$ is constant,
%$(\chi_{q_{\lambda}}(c))_{\overline{\lambda}}\mid_{\lambda=0}=-\overline{A_{c}}$.
We continue to use the symbols defined in the previous sections.
Let $\tilde{g}=4g-2+m$ $(\ge 2)$ be the genus of $\tilde{M}_{q_{0}}$.
Let $\{\alpha_{k},\beta_{k}\}_{k=1}^{\tilde{g}}$ be
a symplectic generator of $H_{1}(\tilde{M}_{q_{0}})$.
Let $x_{\lambda}=\pi(q_{\lambda})$ for instance.
By the bilinear relation,
we have
\begin{align*}
\ext_{x_{\lambda}}(F)
&=\frac{1}{2}\|\omega_{q_{\lambda}}\|^{2}
=\frac{1}{4}\int_{\tilde{M}_{q_{\lambda}}}\omega_{q_{\lambda}}\wedge
{}^{*}\overline{\omega_{q_{\lambda}}}
\\
&=\frac{i}{4}\sum_{k=1}^{\tilde{g}}
\left(
\chi_{q_{\lambda}}(\alpha_{k})\overline{\chi_{q_{\lambda}}(\beta_{k})}
-\chi_{q_{\lambda}}(\beta_{k})\overline{\chi_{q_{\lambda}}(\alpha_{k})}
\right)
\end{align*}
for $\lambda\in D$
(cf. \cite[Corollary in \S III.2.3]{MR1139765}).
From Lemmas \ref{lem:douady-hubbard-local-coordinate}
and \ref{lem:constant-representation},
$\chi_{q_{F,x}}(\alpha_{k})$ and $\chi_{q_{F,x}}(\beta_{k})$
are real analytic  around $x_{0}$ for each $k$,
and hence
so is the extremal length function.
%{\color{red}
%ここではnormは$\int_{X}|q(z)|dxdy$である．
%bilinear relationでは
%$$
%\int \omega\wedge *\overline\psi
%$$
%が周期でかけるのであった．
%}
From Lemma \ref{lem:transversal2},
we have
\begin{align*}
&
\left.
\frac{\partial^{2}}{\partial\lambda\partial\overline{\lambda}}
\left(\chi_{q_{\lambda}}(c_{1})\overline{\chi_{q_{\lambda}}(c_{2})}
\right)
\right|_{\lambda=0}\\
&=
\left.
\frac{\partial^{2}\chi_{q_{\lambda}}(c_{1})}{\partial\lambda\partial\overline{\lambda}}
\right|_{\lambda=0}
\overline{\chi_{q_{0}}(c_{2})}
+\chi_{q_{0}}(c_{1})
\left.
\frac{\partial^{2}\overline{\chi_{q_{\lambda}}(c_{2})}}{\partial\lambda\partial\overline{\lambda}}
\right|_{\lambda=0} \\
&\quad\quad
+\left.
\frac{\partial \chi_{q_{\lambda}}(c_{1})}{\partial\lambda}
\right|_{\lambda=0}
\left.
\frac{\partial \overline{\chi_{q_{\lambda}}(c_{2})}}{\partial\overline{\lambda}}
\right|_{\lambda=0}
+
\left.
\frac{\partial \chi_{q_{\lambda}}(c_{1})}{\partial\overline{\lambda}}
\right|_{\lambda=0}
\left.
\frac{\partial \overline{\chi_{q_{\lambda}}(c_{2})}}{\partial \lambda}
\right|_{\lambda=0}
\\
&=
\left.
\frac{\partial^{2}\chi_{q_{\lambda}}(c_{1})}{\partial\lambda\partial\overline{\lambda}}
\right|_{\lambda=0}
\overline{\chi_{q_{0}}(c_{2})}
+\chi_{q_{0}}(c_{1})
\left.
\frac{\partial^{2}\overline{\chi_{q_{\lambda}}(c_{2})}}{\partial\lambda\partial\overline{\lambda}}
\right|_{\lambda=0} \\
&\quad\quad
+\left.
\frac{\partial \chi_{q_{\lambda}}(c_{1})}{\partial\lambda}
\right|_{\lambda=0}
 \overline{
 \left.
\frac{\partial\chi_{q_{\lambda}(c_{2})}}{\partial\lambda}
\right|_{\lambda=0}
}
+
 \overline{
 \left.
\frac{\partial \chi_{q_{\lambda}}(c_{1})}{\partial\lambda}
\right|_{\lambda=0}
}
\left.
\frac{\partial \chi_{q_{\lambda}}(c_{2})}{\partial \lambda}
\right|_{\lambda=0}
\end{align*}
for $c_{1},c_{2}\in H_{1}(\tilde{M}_{q_{0}})$,
since the real part of $\chi_{q_{\lambda}}(c)$ is constant
for all $c\in H_{1}(\tilde{M}_{q_{0}})$
by Lemma \ref{lem:constant-representation}.
Notice that the sum of the last two terms of above calculation is a real number.
Therefore,
we obtain
\begin{align*}
&
\left.
\frac{\partial^{2}}{\partial\lambda\partial\overline{\lambda}}\left(
\chi_{q_{\lambda}}(\alpha_{k})\overline{\chi_{q_{\lambda}}(\beta_{k})}
-\chi_{q_{\lambda}}(\beta_{k})\overline{\chi_{q_{\lambda}}(\alpha_{k})}
\right)
\right|_{\lambda=0}\\
&=
\int_{\alpha_{k}}\Omega_{0}\overline{\int_{\beta_{k}}\omega_{q_{0}}}
+\overline{\int_{\beta_{k}}\Omega_{0}}
\int_{\alpha_{k}}\omega_{q_{0}}
-
\int_{\beta_{k}}\Omega_{0}\overline{\int_{\alpha_{k}}\omega_{q_{0}}}
-\int_{\alpha_{k}}\overline{\Omega_{0}}
\int_{\beta_{k}}\omega_{q_{0}} \\
&=
\int_{\alpha_{k}}\Omega_{0}\int_{\beta_{k}}\overline{\omega_{q_{0}}}
-
\int_{\beta_{k}}\Omega_{0}\int_{\alpha_{k}}\overline{\omega_{q_{0}}}
+\int_{\beta_{k}}\overline{\Omega_{0}}
\int_{\alpha_{k}}\omega_{q_{0}}
-\int_{\alpha_{k}}\overline{\Omega_{0}}
\int_{\beta_{k}}\omega_{q_{0}}
\end{align*}
where $\Omega_{0}$ is a closed form defined as \eqref{eq:Omega-0-Laplacian}.
Since $\omega_{q_{0}}$ is a holomorphic differential,
${}^{*}\omega_{q_{0}}$
and ${}^{*}\overline{\omega_{q_{0}}}$
are harmonic differentials and they are
perpendicular to the exact and co-exact differentials
in the $L^{2}$-inner product
of differential forms
(cf. \cite[\S II.3]{MR1139765}).
From \eqref{eq:Omega-0-Laplacian},
we obtain
\begin{align*}
\levi{\ext_{\cdot }(F)}[v,\overline{v}]
&=
\left.
\frac{\partial^{2}}{\partial\lambda\partial\overline{\lambda}}
\ext_{x_{\lambda}}(F)
\right|_{\lambda=0}\\
&=
\frac{i}{4}
\sum_{k=1}^{\tilde{g}}
\left(
\int_{\alpha_{k}}\Omega_{0}\int_{\beta_{k}}\overline{\omega_{q_{0}}}
-
\int_{\beta_{k}}\Omega_{0}\int_{\alpha_{k}}\overline{\omega_{q_{0}}}
\right) \\
&\quad
+
\frac{i}{4}
\sum_{k=1}^{\tilde{g}}
\left(
\int_{\beta_{k}}\overline{\Omega_{0}}
\int_{\alpha_{k}}\omega_{q_{0}}
-\int_{\alpha_{k}}\overline{\Omega_{0}}
\int_{\beta_{k}}\omega_{q_{0}}
\right) \\
&=\frac{i}{4}
\int_{\tilde{M}_{q_{0}}}\Omega_{0}\wedge \overline{\omega_{q_{0}}}
-
\frac{i}{4}
\int_{\tilde{M}_{q_{0}}}\overline{\Omega_{0}}\wedge \omega_{q_{0}}
\\
&={\rm Re}
\int_{\tilde{M}_{q_{0}}}
\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
(\pi_{q_{0}})^{*}(\dot{\mu})\omega_{q_{0}}
%dxdy 
\\
&=
2{\rm Re}
\int_{M_{q_{0}}}
\eta_{v}\dot{\mu}
=
2\int_{M_{0}}
\frac{|\eta_{v}|^{2}}{|q_{0}|}
\end{align*}
from the definition of the differential $\eta_{v}$
(cf. \cite[Proposition III.2.3]{MR1139765}).
This implies what we wanted.
\end{proof}

From Theorem \ref{thm:leviform},
we deduce the following inequality.

\begin{corollary}[Gradients and Levi forms]
\label{coro:strong-positivity}
Let $x_{0}=(M_{0},f_{0})\in \teich{g,m}$ and $v=[\dot{\mu}]\in T_{x_{0}}\teich{g,m}$.
Let $F\in \mathcal{MF}$.
Suppose that $q_{0}=q_{F,x_{0}}$ is generic.
Then,
$$
2\left|
(\ext_{\cdot}(F))_{*}[v]
\right|^{2}
\le 
\ext_{x_{0}}(F)\cdot \levi{\ext_{\cdot}(F)}[v,\overline{v}]
$$
holds.
\end{corollary}

\begin{proof}
%Let $x=(M,h)\in \teich{g,m}$ and $v\in T_{x}\teich{g,m}$.
%Let $h\colon \mathbb{D}\to \teich{g,m}$
%be a holomorphic mapping with $h(0)=x$
%and 
%$f_{*}((\partial/\partial\lambda)\mid_{\lambda=0})=v$.
%Let $E(\lambda)=\ext_{f(\lambda)}(F)$.
%\begin{align*}
%\levi{\rho}[v,\overline{v}]
%%&=
%%\frac{
%%(E(0)+a)\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%%-
%%2|\left.E_{\lambda}\right|_{\lambda=0}|^{2}
%%}
%%{(E(0)+a)^{3}}
%%\\
%&=
%\frac{
%c\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%}
%{(E(0)+c)^{3}}
%+
%\frac{
%E(0)\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%2|\left.E_{\lambda}\right|_{\lambda=0}|^{2}
%}
%{(E(0)+c)^{3}}.
%\end{align*}
%For the simplicity,
%we set $a=E(0)$, $b=\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}$
%and $c=E_{\lambda}\right|_{\lambda=0}$.
%From a similar calculation as that given
%around \eqref{eq:Levi-form-comparison},
%Let $x_{0}=(M_{0},f_{0})$.
It follows from Gardiner's formula 
that
\begin{equation}
\label{eq:Gardiner-formula1}
(\ext_{\cdot}(F))_{*}[v]
=
-\int_{M_{0}}\dot{\mu}\,q_{0}
=-\int_{M_{0}}\frac{\overline{\eta_{v}}}{|q_{0}|}q_{0}.
\end{equation}
See \cite[\S11.8]{MR903027}
(See also \S\ref{subsec:Appendix-Gardiner-formula}).
Notice in comparing the Gardiner's original formula
with \eqref{eq:Gardiner-formula1}
that our differential $q_{0}=q_{F,x_{0}}$ has $F$ as the vertical foliation,
while Gardiner considers
the quadratic differential with horizontal foliation $F$.
Hence,
we have to put the minus sign in \eqref{eq:Gardiner-formula1}.
From
Cauchy-Schwarz inequality,
one can see from Theorem \ref{thm:leviform}
that
\begin{align*}
%&E(0)\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%2|\left.E_{\lambda}\right|_{\lambda=0}|^{2} \\
&
\ext_{x_{0}}(F)\levi{\ext_{\cdot}(F)}[v,\overline{v}]
-2\left|
(\ext_{\cdot}(F))_{*}[v]
\right|^{2}
\\
&=
\ext_{x_{0}}(F)\cdot 2
\int_{M_{0}}
\frac{|\eta_{v}|^{2}}{|q_{0}|}
-2\left|
-\int_{M_{0}}\frac{\overline{\eta_{v}}}{|q_{0}|}q_{0}
\right|^{2} \\
&=
2\left(
\int_{M_{0}}
|q_{0}|
\cdot 
\int_{M_{0}}
\frac{|\eta_{v}|^{2}}{|q_{0}|}
-\left|
\int_{M_{0}}\frac{\overline{\eta_{v}}}{|q_{0}|^{1/2}}\frac{q_{0}}{|q_{0}|^{1/2}}
\right|^{2}
\right)
\ge 0,
\end{align*}
which implies what we wanted.
\end{proof}


\subsection{Plurisubharmonicity for subspecies}
\label{subsec:log-plurisubharmonicity}
Let $N$ be a complex manifold.
A function $U$ on $N$ is said to be \emph{plurisubharmonic} if 
$U$ is upper semi-continuous,
and $U\circ f$ is subharmonic for all holomorphic mapping
$f\colon \mathbb{D}\to N$.
A $C^{2}$-function on $N$ is plurisubharmonic
if the Levi form is positive semi-definite.
We call a $C^{2}$-function \emph{strictly plurisubharmonic}
if the Levi form is positive-definite.
When a function $U$ is plurisubharmonic,
so is $g\circ U$
for any increasing convex function $g$ on $\mathbb{R}$
such that the limit $\lim_{t\to -\infty}g(t)$ exists.

In the proof of the following theorem,
we remind that if $U(z)$ is a real-valued positive $C^{2}$-function on a domain
on $\mathbb{C}$,
$\log U$ and $-1/U$
satisfy
\begin{align}
(\log U)_{\lambda\overline{\lambda}}
&=\frac{U\cdot U_{\lambda\overline{\lambda}}-|U_{\lambda}|^{2}}{U^{2}}
\label{eq:log-laplacian}
\\
\left(-\frac{1}{U}\right)_{\lambda\overline{\lambda}}
&=\frac{U\cdot U_{\lambda\overline{\lambda}}-2|U_{\lambda}|^{2}}{U^{3}}.
\label{eq:reciprocal-laplacian}
\end{align}
Especially,
for a positive $C^{2}$-function $U$ on a complex manifold,
if $-1/U$ is plurisubharminic,
$\log U$ and $U$ are also plurisubharmonic.
From \eqref{eq:log-laplacian} and \eqref{eq:reciprocal-laplacian},
Theorem \ref{thm:pluri-subharmonicity} follows from
the following theorem.

\begin{theorem}[Plurisuperharmonicity of reciprocals]
\label{thm:plurisuperharmonicitiy-of-reciprocals}
Let $F_{1},F_{2},\cdots,F_{n}$ be
%essentially complete 
measured foliations.
Let $a_{k},c\ge0$ $(k=1,\cdots,n)$ with $\sum_{k=1}^{n}a_{k}>0$.
If $n\ge 2$,
$a_{k},c>0$
$(k=1,\cdots,n)$
and
 some two of them are transverse
 in the sense of
\eqref{eq:transverse-mf},
the function
$$
\rho(x)=-\frac{1}{c+\sum_{k=1}^{n}a_{k}\ext_{x}(F_{k})}
$$
is a negative,
%real analytic strictly 
plurisubharmonic exhaustion function
of $\teich{g,m}$
with lower bound.
When $n\ge 1$ or $c\ge 0$,
$\rho$ is a negative, 
%real analytic (non-trivial)
plurisubharmonic function
on $\teich{g,m}$
with lower bound.
\end{theorem}

\begin{proof}
We only show the case $n=2$.
The other case can be treated in the same way.
Furthermore,
we may also assume that each $F_{k}$ are essentially complete,
and $F_{1}$ and $F_{2}$ are transverse
in the sense of
\eqref{eq:transverse-mf}.
The general case follows from a standard approximating procedure
(cf. \cite[Theorem 2.6.1]{MR1150978},
\cite[Theorem 2 in Chapter II]{MR0243112} or \cite[3.3 in Chapter II]{MR0344979}).
Under the assumption,
each extremal length function
$\teich{g,m}\ni x\mapsto \ext_{x}(F_{k})$ is 
real analytic.

Let $x=(M,f)\in \teich{g,m}$ and $v\in T_{x_{0}}\teich{g,m}$.
Let $h\colon \mathbb{D}\to \teich{g,m}$
be a holomorphic mapping with $h(0)=x_{0}$
and 
$h_{*}((\partial/\partial\lambda)\mid_{\lambda=0})=v$.
%
%Consider a function on $\teich{g,m}$ defined by
%$$
%\rho(x)=-\frac{1}{a\ext_{x}(F)+b\ext_{x}(G)+c}
%$$
%for $a,b\ge0$ with $a+b>0$ and $c>0$.
Set $E^{1}(\lambda)=\ext_{h(\lambda)}(F_{1})$,
$E^{2}(\lambda)=\ext_{h(\lambda)}(F_{2})$
and $E=a_{1}E^{1}+a_{2}E^{2}$ for simplicity.

Then,
the Levi form of $\rho$ is 
\begin{align*}
\levi{\rho}[v,\overline{v}]
%&=
%\frac{
%(E(0)+a)\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%2|\left.E_{\lambda}\right|_{\lambda=0}|^{2}
%}
%{(E(0)+a)^{3}}
%\\
&=
\frac{
c\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
}
{(c+E(0))^{3}}
+
\frac{
E(0)\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
-
2|\left.E_{\lambda}\right|_{\lambda=0}|^{2}
}
{(c+E(0))^{3}}.
\end{align*}
%For the simplicity,
%we set $a=E(0)$, $b=\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}$
%and $c=E_{\lambda}\right|_{\lambda=0}$.
%From a similar calculation as that given
%around \eqref{eq:Levi-form-comparison},
%Applying Gardiner's formula
%and Cauchy-Schwarz inequality,
%one can see from Theorem \ref{thm:leviform}
%that
%From Lemma \ref{coro:strong-positivity},
%we have
%\begin{align*}
%&E(0)\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%2|\left.E_{\lambda}\right|_{\lambda=0}|^{2} \\
%&=
%\ext_{x_{0}}(F_{k})\cdot 2
%\int_{M_{0}}
%\frac{|\eta_{v}|^{2}}{|q_{F_{k},x_{0}}|}
%-2\left|
%\int_{M_{0}}\frac{\overline{\eta_{v}}}{|q_{F_{k},x_{0}}|}q_{F_{k},x_{0}}
%\right| \\
%&=
%2\left(
%\int_{M_{0}}
%|q_{F_{k},x_{0}}|
%\cdot 
%\int_{M_{0}}
%\frac{|\eta_{v}|^{2}}{|q_{F_{k},x_{0}}|}
%-\left|
%\int_{M_{0}}\frac{\overline{\eta_{v}}}{|q_{F_{k},x_{0}}|^{1/2}}\frac{q_{F_{k},x_{0}}}{|q_{F_{k},x_{0}}|^{1/2}}
%\right|
%\right) \\
%&\ge 0
%\end{align*}
%for each $k=1,2$,
%where $x_{0}=(M_{0},f_{0})$.
%See \cite[\S11.8]{MR903027}.
%See also \S\ref{subsec:Appendix-Gardiner-formula}.
Therefore,
we conclude 
from Corollary \ref{coro:strong-positivity}
that
\begin{align*}
&E(0)\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
-
2|\left.E_{\lambda}\right|_{\lambda=0}|^{2} \\
&=
a_{1}^{2}\left(
E^{1}(0)\left.E^{1}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
-
2|\left.E^{1}_{\lambda}\right|_{\lambda=0}|^{2}
\right)
+a_{2}^{2}\left(
E^{2}(0)\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
-
2|\left.E^{2}_{\lambda}\right|_{\lambda=0}|^{2}
\right) \\
&\quad+
a_{1}a_{2}\left(
E^{1}(0)\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}+
E^{2}(0)\left.E^{1}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
-
4{\rm Re}\left(
\overline{\left.E^{1}_{\lambda}\right|_{\lambda=0}}
\left.E^{2}_{\lambda}\right|_{\lambda=0}
\right)
%-
%2\overline{\left.E^{2}_{\lambda}\right|_{\lambda=0}}
%\left.E^{1}_{\lambda}\right|_{\lambda=0}
\right) \\
&\ge
%(E^{1}(0)\left.E^{1}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%2|\left.E^{1}_{\lambda}\right|_{\lambda=0}|^{2})
%+(E^{2}(0)\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%2|\left.E^{2}_{\lambda}\right|_{\lambda=0}|^{2}) \\
%&\quad+
a_{1}a_{2}\left(
E^{1}(0)\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}+
E^{2}(0)\left.E^{1}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
-
4
\left|
\left.E^{1}_{\lambda}\right|_{\lambda=0}
\right|
\left|
\left.E^{2}_{\lambda}\right|_{\lambda=0}
\right|
\right) \\
&\ge
a_{1}a_{2}\left(
E^{1}(0)\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}+
E^{2}(0)\left.E^{1}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
\right.
\\
&
\left.\qquad-
2
E^{1}(0)^{1/2}
\left(
\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
\right)^{1/2}
E^{1}(0)^{1/2}
\left(\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
\right)^{1/2}
\right) \\
&=
a_{1}a_{2}\left(
E^{1}(0)^{1/2}
\left(\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
\right)^{1/2}
-
E^{1}(0)^{1/2}
\left(
\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
\right)^{1/2}
\right)^{2}
\ge 0
\end{align*}
and hence
$$
\levi{\rho}[v,\overline{v}]
\ge
\frac{
c\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
}
{(c+E(0))^{3}}
=
c\frac{
a_{1}^{2}\levi{\ext_{\cdot}(F_{1})}[v,\overline{v}]+a_{2}^{2}\levi{\ext_{\cdot}(F_{2})}[v,\overline{v}]
}
{(c+a_{1}\ext_{x_{0}}(F_{1})+a_{2}\ext_{x_{0}}(F_{2}))^{3}}
$$
when 
%$v\ne 0$
%and 
$a+b>0$, $c>0$.
Indeed,
since $q_{F,x}$ is generic,
the anti-complex linear map
$$
T_{x_{0}}\teich{g,m}\ni v\mapsto \eta_{v}\in \quaddiff{x_{0}}
$$
is an isomorphism.
%This means that the Levi form of $\rho$
%is positive for all non-zero tangent vectors.
Therefore,
$\rho$ is (real analytic) strictly plurisubharmonic on $\teich{g,m}$.
From the definition,
$\rho$ satisfies that
$\rho(x)<0$ for $x\in \teich{g,m}$.
Since $F_{1}$ and $F_{2}$ are transverse,
for any $\epsilon>0$,
there is a compact set $K\subset \teich{g,m}$
such that $\rho>-\epsilon$ for all $x\in \teich{g,m}-K$
(cf. \S\ref{Kerckhoff-Minsky}).
Since $\rho(x)>-1/a$ for all $x\in \teich{g,m}$,
%$\rho$ attains a minimum at some point in $\teich{g,m}$.
%This means
% that $C_{0}^{-1}=|\min_{x\in \teich{g,m}}\rho(x)|$
%exists and is positive.
the function $\rho(x)$
satisfies the desired condition.
%so that $\teich{g,m}$ is hyperconvex.
\end{proof}

%
%\begin{proof}[Proof of Theorem \ref{thm:pluri-subharmonicity}]
%Let $F\in \mathcal{MF}$.
%Suppose first that
%$F$ is essentially complete.
%%has only three prong singularities one prong singularities at punctures,
%%and does not contain saddle connections.
%It is known that such measured foliations consists of a dense subset in $\mathcal{MF}$
%(cf. \cite[Proposition 9.5.11]{Thuston-LectureNote}).
%In this case,
%the Hubbard-Masur differential $q_{F,x}$
%is generic
%for all $x=(M,f)\in \teich{g,m}$.
%
%We use the same symbols given in Theorem \ref{thm:leviform}.
%Gardiner's differential formula tells us that
%$$
%\frac{\partial}{\partial \lambda}\ext_{x_{\lambda}}(F)
%=\int_{M}\dot{\mu}q_{F,x}
%=\int_{M}\frac{\overline{\eta_{v}}}{|q_{F,x}|}q_{F,x}.
%$$
%See \cite[\S11.8]{MR903027}
%(See also \S\ref{subsec:Appendix-Gardiner-formula} for an alternative approach).
%By Theorem \ref{thm:leviform} and \eqref{eq:log-laplacian},
%we have
%\begin{align}
%&\levi{\log\ext_{\cdot }(F)}[v,\overline{v}]=
%\left.
%\frac{\partial^{2}}{\partial\lambda\partial\overline{\lambda}}
%\log\ext_{x_{\lambda}}(F)
%\right|_{\lambda=0}
%\nonumber
%\\
%&=
%\frac{1}{\ext_{x_{0}}(F)^{2}}
%\left(
%\ext_{x_{0}}(F)
%\left.
%\frac{\partial^{2}\ext_{x_{\lambda}}(F)}{\partial\lambda\partial\overline{\lambda}}
%\right|_{\lambda=0}
%-
%\left|
%\left.
%\frac{\partial\ext_{x_{\lambda}}(F)}{\partial\lambda}
%\right|_{\lambda=0}
%\right|^{2}
%\right)
%\nonumber
%\\
%&=
%\frac{1}{\|q_{F,x_{0}}\|^{2}}
%\left(
%2\int_{M}|q_{F,x_{0}}|
%\int_{M}\frac{|\eta_{v}|^{2}}{|q_{F,x_{0}}|}
%-
%\left|
%\int_{M}\frac{\overline{\eta_{v}}}{|q_{F,x_{0}}|}q_{F,x_{0}}
%\right|^{2}
%\right) 
%\nonumber
%\\
%&\ge 
%\frac{1}{\|q_{F,x_{0}}\|}\int_{M}\frac{|\eta_{v}|^{2}}{|q_{F,x_{0}}|}
%%\nonumber
%%\\
%%&=\frac{1}{2}\levi{\log\ext_{\cdot }(F)}[v,\overline{v}]
%\label{eq:Levi-form-comparison}
%\end{align}
%by Cauchy-Schwarz inequality.
%Since $q_{F,x}$ is generic,
%the anti-complex linear map
%$$
%T_{x}\teich{g,m}\ni v\mapsto \eta_{v}\in \quaddiff{x}
%$$
%is an isomorphism.
%This means that the Levi form of the log-extremal length funciton
%is positive for all non-zero tangent vectors,
%and
%the log-extremal length function $\teich{g,m}\ni x\mapsto \log\ext_{x}(F)$
%is strictly plurisubharmonic.
%The plurisubharomnicity for general measured foliations
%follows from approximation.
%\end{proof}

%From the proof of Theorem \ref{thm:pluri-subharmonicity},
%Lemma \ref {coro:strong-positivity},
We also obtain the following comparizon.

%\begin{corollary}[Positibity of Levi form of log-ext]
%\label{coro:strong-positivity2}
%Let $x_{0}\in \teich{g,m}$ and $v=[\dot{\mu}]\in T_{x_{0}}\teich{g,m}$.
%Let $F\in \mathcal{MF}$.
%Suppose that $q_{0}=q_{F,x_{0}}$ is generic.
%Then,
%$$
%\frac{1}{\ext_{x_{0}}(F)^{2}}\left|
%(\ext_{\cdot}(F))_{*}[v]
%\right|^{2}
%\le 
%\levi{\log\ext_{\cdot}(F)}[v,\overline{v}]
%$$
%holds.
%\end{corollary}

\begin{theorem}[Gradients and Levi forms for log-extremal length functions]
\label{thm:strong-positivity2-2}
For any measured foliation
$F$,
%and $v=[\dot{\mu}]\in T_{x_{0}}\teich{g,m}$.
%Let $F\in \mathcal{MF}$.
%Suppose that $q_{0}=q_{F,x_{0}}$ is generic.
\begin{align*}
%\frac{2\pi }{\ext_{x_{0}}(F)^{2}}
%\sqrt{-1}\left(
%\partial 
d\log \ext_{\cdot}(F)
\wedge
%\overline{\partial} 
d^{c}\log \ext_{\cdot}(F)
%\right)
&\le 
\frac{1}{2\ext_{\cdot}(F)}
%\sqrt{-1}\partial\overline{\partial}
dd^{c}
\,
\ext_{\cdot}(F)
%\label{eq:gradient-log}
%\\
%&
\le 
%\sqrt{-1}\partial\overline{\partial}
dd^{c}\log\ext_{\cdot}(F)
%\label{eq:gradient-log2}
%\\
%&\le 
%\frac{1}{\ext_{\cdot}(F)}\sqrt{-1}\partial\overline{\partial}\,\ext_{\cdot}(F)
%\label{eq:gradient-log3}
\end{align*}
holds
in the sense of currents,
where $d=\partial+\overline{\partial}$
and $d^{c}=i(\overline{\partial}-\partial)$.
Especially,
the log-extremal length function
$\log\ext_{\cdot}(F)$ is a real analytic strictly plurisubharmonic function on $\teich{g,m}$,
when $F$ is essentially complete.
\end{theorem}

\begin{proof}
By approximation,
we may assume that $F$ is essentially complete.
Then,
the inequalities follow
from the combinations of \eqref{eq:log-laplacian}
%\eqref{eq:reciprocal-laplacian},
and
Corollary \ref{coro:strong-positivity}.
%$$
%\left|
%(\log \ext_{\cdot}(F))_{*}[v]
%\right|^{2}
%\le 
%\levi{\log\ext_{\cdot}(F)}[v,\overline{v}]
%\quad (v\in T\teich{g,m}),
%$$
%which
%follows from the calculation
%in the proof of Theorem \ref{thm:plurisuperharmonicitiy-of-reciprocals}.
\end{proof}

%\begin{corollary}
%When
%$F\in \mathcal{MF}$ has only three prong singularities and does not contain saddle connections,
%the log-extremal length function $\log\ext_{\cdot}(F)$ is
%real analytic and strictly log-plurisubharmonic funciton.
%\end{corollary}
%
%\begin{proof}
%Let $x\in \teich{g,m}$.
%For $v\in T_{x}\teich{g,m}$,
%we define $\eta_{v}\in \quaddiff{x}$ as Lemma \ref{thm:leviform}.
%Since $q_{F,x}$ is generic,
%$T_{x}\teich{g,m}\ni v\mapsto \eta_{v}\in \quaddiff{x}$ is an isomorphism.
%This means that the Levi form is positive for all non-zero tangent vectors.
%\end{proof}

%
%\subsection{Plurisuperharmonicity of reciprocals}
%\label{subsec:hyperconvexity}
%To prove Corollary \ref{coro:hyperconvexity},
%we claim
%the following.
%%\begin{theorem}[Plurisuperharmonicity of reciprocals]
%\label{thm:plurisuperharmonicitiy-of-reciprocals}
%Let $F$ and $G$ be two essentially complete measured foliations.
%Suppose that $F$ and $G$ are transverse in the sense of
%\eqref{eq:transverse-mf}.
%Let $a,b,c\ge0$ with $a+b>0$.
%Then,
%the function
%$$
%\rho(x)=-\frac{1}{a\ext_{x}(F)+b\ext_{x}(G)+c}
%$$
%is a negative, real analytic strictly plurisubharmonic exhaustion function
%of $\teich{g,m}$
%with lower bound
%when $a,b,c>0$.
%When $c=0$ but $a+b>0$,
%$\rho$ is a negative, real analytic (non-trivial)
%plurisubharmonic function
%on $\teich{g,m}$.
%\end{theorem}
%
%\begin{proof}
%Let $x\in \teich{g,m}$ and $v\in T_{x}\teich{g,m}$.
%Let $f\colon \mathbb{D}\to \teich{g,m}$
%be a holomorphic mapping with $f(0)=x$
%and 
%$f_{*}((\partial/\partial\lambda)\mid_{\lambda=0})=v$.
%%
%%Consider a function on $\teich{g,m}$ defined by
%%$$
%%\rho(x)=-\frac{1}{a\ext_{x}(F)+b\ext_{x}(G)+c}
%%$$
%%for $a,b\ge0$ with $a+b>0$ and $c>0$.
%Set $E^{1}(\lambda)=\ext_{f(\lambda)}(F)$,
%$E^{2}(\lambda)=\ext_{f(\lambda)}(G)$
%and $E=aE^{1}+bE^{2}$ for simplicity.
%
%Then,
%the Levi form of $\rho$ is 
%\begin{align*}
%\levi{\rho}[v,\overline{v}]
%%&=
%%\frac{
%%(E(0)+a)\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%%-
%%2|\left.E_{\lambda}\right|_{\lambda=0}|^{2}
%%}
%%{(E(0)+a)^{3}}
%%\\
%&=
%\frac{
%c\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%}
%{(E(0)+c)^{3}}
%+
%\frac{
%E(0)\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%2|\left.E_{\lambda}\right|_{\lambda=0}|^{2}
%}
%{(E(0)+c)^{3}}.
%\end{align*}
%%For the simplicity,
%%we set $a=E(0)$, $b=\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}$
%%and $c=E_{\lambda}\right|_{\lambda=0}$.
%From a similar calculation as that given
%around \eqref{eq:Levi-form-comparison},
%from Cauchy-Schwarz inequality,
%one can see that
%$$
%E^{k}(0)\left.E^{k}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%2|\left.E^{k}_{\lambda}\right|_{\lambda=0}|^{2}\ge 0
%$$
%for each $k=1,2$.
%Therefore,
%we conclude that
%\begin{align*}
%&E(0)\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%2|\left.E_{\lambda}\right|_{\lambda=0}|^{2} \\
%&=
%a^{2}\left(
%E^{1}(0)\left.E^{1}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%2|\left.E^{1}_{\lambda}\right|_{\lambda=0}|^{2}
%\right)
%+b^{2}\left(
%E^{2}(0)\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%2|\left.E^{2}_{\lambda}\right|_{\lambda=0}|^{2}
%\right) \\
%&\quad+
%ab\left(
%E^{1}(0)\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}+
%E^{2}(0)\left.E^{1}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%4{\rm Re}\left(
%\overline{\left.E^{1}_{\lambda}\right|_{\lambda=0}}
%\left.E^{2}_{\lambda}\right|_{\lambda=0}
%\right)
%%-
%%2\overline{\left.E^{2}_{\lambda}\right|_{\lambda=0}}
%%\left.E^{1}_{\lambda}\right|_{\lambda=0}
%\right) \\
%&\ge
%%(E^{1}(0)\left.E^{1}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%%-
%%2|\left.E^{1}_{\lambda}\right|_{\lambda=0}|^{2})
%%+(E^{2}(0)\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%%-
%%2|\left.E^{2}_{\lambda}\right|_{\lambda=0}|^{2}) \\
%%&\quad+
%ab\left(
%E^{1}(0)\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}+
%E^{2}(0)\left.E^{1}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%-
%4
%\left|
%\left.E^{1}_{\lambda}\right|_{\lambda=0}
%\right|
%\left|
%\left.E^{2}_{\lambda}\right|_{\lambda=0}
%\right|
%\right) \\
%&\ge
%ab\left(
%E^{1}(0)\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}+
%E^{2}(0)\left.E^{1}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%\right.
%\\
%&
%\left.\qquad-
%2
%E^{1}(0)^{1/2}
%\left(
%\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%\right)^{1/2}
%E^{1}(0)^{1/2}
%\left(\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%\right)^{1/2}
%\right) \\
%&=
%ab\left(
%E^{1}(0)^{1/2}
%\left(\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%\right)^{1/2}
%-
%E^{1}(0)^{1/2}
%\left(
%\left.E^{2}_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%\right)^{1/2}
%\right)^{2}
%\ge 0
%\end{align*}
%and hence
%$$
%\levi{\rho}[v,\overline{v}]
%\ge
%\frac{
%c\left.E_{\lambda\overline{\lambda}}\right|_{\lambda=0}
%}
%{(E(0)+c)^{3}}
%=
%c\frac{
%a^{2}\levi{\ext_{\cdot}(F)}[v,\overline{v}]+b^{2}\levi{\ext_{\cdot}(G)}[v,\overline{v}]
%}
%{(a\ext_{x}(F)+\ext_{x}(G)+c)^{3}}
%>0
%$$
%when $v\ne 0$
%and $a+b>0$, $c>0$
%from Theorem \ref{thm:pluri-subharmonicity}.
%Therefore,
%$\rho$ is (real analytic) strictly plurisubharmonic on $\teich{g,m}$.
%From the definition,
%$\rho$ satisfies that
%$\rho(x)<0$
%and for any $\epsilon>0$,
%there is a compact set $K\subset \teich{g,m}$
%such that $\rho>-\epsilon$ for all $x\in \teich{g,m}-K$
%(cf. \S\ref{Kerckhoff-Minsky}).
%Since $\rho(x)>-1/a$ for all $x\in \teich{g,m}$,
%%$\rho$ attains a minimum at some point in $\teich{g,m}$.
%%This means
%% that $C_{0}^{-1}=|\min_{x\in \teich{g,m}}\rho(x)|$
%%exists and is positive.
%the function $\rho(x)$
%satisfies the desired condition.
%%so that $\teich{g,m}$ is hyperconvex.
%\end{proof}


\section{Gemetric interpretation of $\eta_{v}$}
\label{sec:geometric-interpretation}
\subsection{Motivation}
Let $x_{0}=(M_{0},f_{0})\in \teich{g,m}$.
Suppose $q_{0}=q_{F,x_{0}}$ is generic.
%In \citeMR1099913,
%Gardiner and Masur
%observed that
%the mapping
%\begin{equation}
%\label{eq:Gardiner-Masur}
%\GM\colon
%\quaddiff{g,m}\ni q\mapsto (\horizontal(q),\vertical(q))\in \mathcal{MF}\times \mathcal{MF}
%\end{equation}
%is injective,
%where $\horizontal(q)$ is the horizontal foliation of $q$.
%Namely,
%$\horizontal(q)=\vertical(-q)$.
%Let $\xi_{F,G}=\GM^{-1}(F,G)$
%for $(F,G)\in \GM(\quaddiff{g,m})$.
Let $v=[\dot{\mu}]\in T_{x_{0}}\teich{g,m}$.
We defined the quadratic differential $\eta_{v}\in \quaddiff{g,m}$
to satisfy
\begin{equation}
\label{eq:pairing-and-projection}
\langle v,\psi\rangle=\int_{M_{0}}\frac{\overline{\eta_{v}}}{|q_{0}|}\psi
\end{equation}
for all $\psi\in \quaddiff{g,m}$
in Lemma \ref{lem:transversal} and Theorem \ref{thm:leviform}.
The definition of $\eta_{v}$
implies
as if the infinitesimal Beltrami differential $\dot{\mu}$
were infinitesimally equivalent to the ``infinitesimal deformation'' along
$\overline{\eta_{v}}/|q_{0}|$
on $M_{q_{0}}$
(cf. \cite{MR1215481}).
However,
$\overline{\eta_{v}}/|q_{0}|$
is not a Beltrami differential in general,
becauce it may have $1/|z|$-singularities at zeroes of $q_{0}$
and hence it could happen $\|\overline{\eta_{v}}/|q_{0}|\|_{\infty}=\infty$.
Therefore,
the above discussion does not interpret correctly
the geometry of the anti-complex linear map
\begin{equation}
\label{eq:corresponding-v-eta}
T_{x_{0}}\teich{g,m}\ni v\mapsto \eta_{v}\in \quaddiff{x_{0}}
\end{equation}
 in general.
In this section,
we shall try to give a geometric description of the correspondence
\eqref{eq:corresponding-v-eta}.

\subsection{Geometric interpretation}
From \cite[Theorem 5.8]{2012arXiv1203.0273D} and \cite{MR1099913},
the map 
\begin{equation}
\label{eq:Gardiner-Masur}
%\GM\colon
\quaddiff{g,m}\ni q\mapsto (\horizontal(q),\vertical(q))\in \mathcal{MF}\times \mathcal{MF}
\end{equation}
is a real analytic diffeomorphism
around $q_{0}$.
%
%Therefore,
%we can obtain a real-analytic diffeomorphism
Hence, the map
\begin{equation}
\label{eq:Gardiner-Masur2}
\complexstructure\colon
\teich{g,m}\times \teich{g,m}
\ni (x,y)\mapsto -q_{\horizontal(q_{F,x}),y}\in 
\quaddiff{g,m}
\end{equation}
is also a real analytic diffeomorphism on a neighborhood $N_{1}\times N_{1}$
of $(x_{0},x_{0})$
onto the image,
where $N_{1}$ is a neighborhood of $x_{0}=\pi(q_{0})\in \teich{g,m}$.
Notice that 
$\horizontal(\complexstructure(x,y))=\horizontal(q_{F,x})$,
$\complexstructure(x,y)\in \quaddiff{y}$
and $\complexstructure(x,x)=q_{F,x}$ for all $x,y\in N_{1}$.
Let $\complexstructure_{y}(x)=\complexstructure(x,y)$ for $(x,y)\in N_{1}\times N_{1}$
(Figure \ref{fig:mapJ}).
\begin{figure}
\includegraphics[height=6cm]{mapJ.eps}
\caption{A schematic of the map $\mathcal{J}_{x_{0}}$.}
\label{fig:mapJ}
\end{figure}


%for $x,y\in N_{1}$,
%the derivative of the map \eqref{eq:Gardiner-Masur2}
%along the first coordinate gives a linear mapping
%$$
%L_{(x,y)}\colon T_{x}\teich{g,m}\to \quaddiff{y}
%\cong
%{\rm Ker}(\pi_{*}\mid_{\Xi(x,y)}).
%$$

\begin{theorem}[Geometric interpretation]
\label{thm:duality}
For $v\in T_{x_{0}}\teich{g,m}$,
we have
$$
(\complexstructure_{x_{0}})_{*}[v]=-4\eta_{v},
$$
where $\eta_{v}\in \quaddiff{x_{0}}$ is taken
as \eqref{eq:pairing-and-projection}.
\end{theorem}

\begin{proof}
Consider the triangulation $\Delta_{q}$ and
the train track $\tau_{q_{0}}$ on $M_{q}$
for the horizontal foliation $|{\rm Im}(\sqrt{q})|$
of $q$
%as the proof of Lemma \ref{lem:constant-representation}
(cf. \cite[Theorem 5.8]{2012arXiv1203.0273D}).
We may assume that $\Delta_{q}$ contains no horizontal edge
(cf. \cite[Lemma 5.7]{2012arXiv1203.0273D}).
Consider the lifts $\tilde{\Delta}_{q}$ and $\tilde{\tau}_{q}$ of
$\Delta_{q}$ and $\tau_{q}$ to $\tilde{M}_{q}$,
respectivey.
We orient each edge of $\tilde{\Delta}_{q}$ so that
${\rm Im}(\omega_{q})$ is positive along the edge.
Each branch $b$ of $\tilde{\tau}_{q}$ intersects
a unique edge $e_{b}$ of $\tilde{\Delta}_{q}$.
We orient $b$ so that $b\cdot e_{b}=+1$,
where $\cdot$ means the algebraic intersection number.
By taking $U$ sufficiently small,
we may also assume that the marking $\tilde{M}_{q_{0}}\to \tilde{M}_{q}$
induces isomorphisms from $\tilde{\Delta}_{q_{0}}$ and
$\tilde{\tau}_{q_{0}}$
to $\tilde{\Delta}_{q}$ and $\tilde{\tau}_{q}$,
respectively.

From \cite[Theorem 5.8]{2012arXiv1203.0273D},
the tangential map of
the map $\horizontal\colon \quaddiff{x_{0}}\to \mathcal{MF}$
of the horizontal foliations
satisfies 
\begin{equation}
\label{eq:Dumas1}
%\frac{1}{2}
%{\rm Im}\int_{\tilde{M}_{q_{0}}}
%\frac{(\pi_{q_{0}})^{*}(\psi_{1})\overline{(\pi_{q_{0}})^{*}(\psi_{2})}}{4|\omega_{q_{0}}|^{2}}=
{\rm Im}\int_{M_{q_{0}}}\frac{\psi_{1}\overline{\psi_{2}}}{4|q_{0}|}
=\ThurstonSymplectic{\horizontal_{*}(\psi_{1})}{\horizontal_{*}(\psi_{2})}{\tau_{q_{0}}}
\end{equation}
for $\psi_{1},\psi_{2}\in 
T_{q_{0}}\quaddiff{x_{0}}\cong \quaddiff{x_{0}}$,
where $\ThurstonSymplectic{\cdot}{\cdot}{\tau_{q_{0}}}$
is the Thurston's symplectic form on $W(\tau_{q_{0}})$,
where $W(\tau_{q_{0}})$ is the space
of real valued functions on the branches of $\tau_{q_{0}}$
satisfying the switch condition.
It is known that the space $W(\tau_{q_{0}})$
is identified with the tangent space of $\mathcal{MF}$
at $\horizontal(q_{0})$
since $\tau_{q_{0}}$ is maximal
(cf. \cite[\S3.2]{MR1144770}).
%We can treat in the same way even if $\tilde{\tau'_{q_{0}}$ is a bigon track
%(cf. \cite[\S3.4]{MR1144770}).

From the definition,
$|{\rm Im}(\omega_{q_{F,x}})|$ is the lift of the horizontal foliation
of $q_{F,x}$ to $\tilde{M}_{q_{F,x}}$.
By the definition of the orientation of edges of $\tilde{\Delta}_{q_{0}}$,
the image of the map
$N_{1}\ni x\mapsto \horizontal(q_{F,x})\in \mathcal{MF}$
at $x$ is identified with $w_{x}\in W(\tilde{\tau}_{q_{0}})$
which is defined by
$$
w_{x}(b)=\int_{b}|{\rm Im}(\omega_{q_{F,x}})|
=\int_{b}{\rm Im}(\omega_{q_{F,x}}).
$$
From Lemmas \ref{lem:Douady-Hubbard} and \ref{lem:transversal},
the image of $v\in T_{x_{0}}\teich{g,m}$ under the differential of the map
$N_{1}\ni x\mapsto {\rm Im}(\chi_{q_{F,x}})$
is identified with
the homomorphism
\begin{equation}
\label{eq:derivative1}
V_{v}
\colon H_{1}(\tilde{M}_{q_{0}})
\ni c\mapsto 
-2{\rm Im}\int_{c}
\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
\in \mathbb{R}.
\end{equation}
Indeed,
when $\{q_{\lambda}\}_{\lambda}\in D$
is taken as Lemma \ref{lem:transversal},
\eqref{eq:derivative1}
is derived from
\begin{align*}
\frac{\partial {\rm Im}(\chi_{q_{\lambda}}(c))}{\partial t}
&=
-i\left(\frac{\partial \chi_{q_{\lambda}}(c)}{\partial \lambda}
+\frac{\partial \chi_{q_{\lambda}}(c)}{\partial \overline{\lambda}}
\right)
\\
&=-i\int_{c}
\overline{
\left(
\frac{(\pi_{q_{\lambda}})^{*}(\eta_{\lambda})}{\omega_{q_{\lambda}}}
\right)}
-\frac{(\pi_{q_{\lambda}})^{*}(\eta_{\lambda})}{\omega_{q_{\lambda}}}
\\
&=-2{\rm Im}\int_{c}
\frac{(\pi_{q_{\lambda}})^{*}(\eta_{\lambda})}{\omega_{q_{\lambda}}}
\end{align*}
where $\lambda=t+is$,
since the real part of $\chi_{q_{\lambda}}$ is constant on $D$.

Let $p\colon \tilde{\tau}_{q_{0}}\to \tau_{q_{0}}$ be the
induced covering.
For a weight $w\in W(\tau_{q_{0}})$,
we define $C_{w}\in H_{1}(\tilde{M}_{q_{0}},\mathbb{R})^{-}$
as the homology class of a cycle
$$
\sum_{\mbox{$b$:branches of $\tilde{\tau}_{q_{0}}$}}w(p(b))b.
$$
%Let $\sigma(c)=\sigma_{c}(z)dz+\overline{\sigma_{c}(z)}d\overline{z}$
%be the reproducing harmonic differential on $\tilde{M}_{q_{0}}$
%associated to $c\in H_{1}(\tilde{M}_{q_{0}})$,
%where $\sigma_{c}(z)dz$ is a holomorphic $1$-form on $\tilde{M}_{q_{0}}$
%(we call this holomorphic $1$-form the \emph{holomorphic part} of $\sigma(c)$).
%Namely,
%the differential $\sigma(c)$ is a unique harmonic differential on $\tilde{M}_{q_{0}}$
%which satisfies
%$$
%\int_{c}\omega=\int_{\tilde{M}_{q_{0}}}\omega\wedge {}^{*}\overline{\sigma(c)}
%$$
%for all closed $1$-form $\omega$ on $\tilde{M}_{q_{0}}$.
Then,
\begin{align*}
w_{x}(C_{w})
&=
\sum_{b}w(p(b))w_{x}(b)
\\
V_{v}(C_{w})
&=
-2{\rm Im}\int_{C_{w}}
\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
=
-2
\sum_{b}w(p(b))
{\rm Im}\int_{b}
\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}.
%=
%2{\rm Im}\int_{\tilde{M}_{q_{0}}}
%\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
%\omega\wedge {}^{*}\overline{\sigma(C_{w})}
\end{align*}
These equations imply that
the differential of the map $N_{1}\ni x\mapsto w_{x}\in W(\tilde{\tau}_{q_{0}})$ at $x=x_{0}$ is equal to the weight
(which we abbreviate as)
$$
V_{v}(b)=-2\,{\rm Im}\int_{b}
\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}.
$$
The weight $V_{v}$ is invariant under the action of covering
transformation $r_{0}$ of $\tilde{M}_{q_{0}}\to M_{q_{0}}$
on $\tilde{\tau}_{q_{0}}$,
because
for any branch $b$ of $\tilde{\tau}_{q_{0}}$,
the orientation of $r_{0}(b)$ is opposite to the orientation induced
from $b$.
Thus,
$V_{v}$ descends to a weight $p_{*}(V_{v})$
on $\tau_{q_{0}}$.

Let $\psi_{0}=(\complexstructure_{x_{0}})_{*}[v]$.
For all $\psi\in \quaddiff{x_{0}}\subset T_{q_{0}}\quaddiff{g,m}$,
we define $w(\psi)\in W(\tilde{\tau}_{q_{0}})$ by
$$
w(\psi)(b)={\rm Im}\int_{b}\frac{(\pi_{q_{0}})^{*}(\psi)}{2\omega_{q_{0}}}.
$$
Then,
we have
$\horizontal_{*}[\psi]=p_{*}(w(\psi))$
(cf. \cite[Proof of Theorem 5.8]{2012arXiv1203.0273D}).
By the definition of $\complexstructure_{x_{0}}$,
$\horizontal\circ \complexstructure_{x_{0}}(x)=\horizontal(q_{F,x})$
for all $x\in N_{1}$.
Hence,
$\horizontal_{*}[\psi_{0}]=p_{*}(V_{v})$
as elements of $W(\tilde{\tau}_{q_{0}})
\cong T_{\horizontal(q_{0})}\mathcal{MF}$.
From \eqref{eq:Dumas1}
and \cite[(5.4)]{2012arXiv1203.0273D},
the following holds for all $\psi\in \quaddiff{x_{0}}$:
\begin{align*}
{\rm Im}\int_{M_{q_{0}}}\frac{\psi_{0}\overline{\psi}}{4|q_{0}|}
&=\frac{1}{2}\,{\rm Im}\int_{\tilde{M}_{q_{0}}}
\frac{(\pi_{q_{0}})^{*}(\psi_{0})\overline{(\pi_{q_{0}})^{*}(\psi)}}
{4|\omega_{q_{0}}|^{2}}
\\
&=
\frac{1}{2}\,\ThurstonSymplectic{V_{v}}{w(\psi)}{\tilde{\tau}_{q_{0}}} \\
&=
-2\int_{\tilde{M}_{q_{0}}}
{\rm Im}\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{2\omega_{q_{0}}}
\wedge
{\rm Im}\frac{(\pi_{q_{0}})^{*}(\psi)}{2\omega_{q_{0}}} \\
&=
-{\rm Re}\int_{\tilde{M}_{q_{0}}}
\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{2\omega_{q_{0}}}
\wedge
\overline{\frac{(\pi_{q_{0}})^{*}(\psi)}{2\omega_{q_{0}}}}
\\
&=
-2{\rm Im}\int_{\tilde{M}_{q_{0}}}
\frac{(\pi_{q_{0}})^{*}(\eta_{v})\overline{(\pi_{q_{0}})^{*}(\psi)}}
{4|\omega_{q_{0}}|^{2}}
=
-{\rm Im}\int_{M_{q_{0}}}\frac{\eta_{v}\overline{\psi}}{|q_{0}|}.
\end{align*}
Therefore,
we have
$(\complexstructure_{x_{0}})_{*}[v]
=\psi_{0}=-4\eta_{v}$.
%%which is defined by
%%$$
%%%\frac{\partial \chi_{q_{\lambda}}(c)}{\partial t}
%%%=
%%%\frac{\partial \chi_{q_{\lambda}}(c)}{\partial \lambda}
%%%+
%%%\frac{\partial \chi_{q_{\lambda}}(c)}{\partial \overline{\lambda}}
%%V_{v}(c)=
%%2i{\rm Im}\int_{c}
%%\frac{(\pi_{q_{0}})^{*}(\psi_{1})}{\omega_{q_{0}}}
%%$$
%for $c\in H_{1}(\tilde{M}_{q_{0}})$,
%where $\lambda=t+is$.
\end{proof}

\section{Appendix}


%\section{Appendix}
\subsection{Alternative approach to Gardiner's formula}
\label{subsec:Appendix-Gardiner-formula}
Applying our method in Theorem \ref{thm:leviform},
one can also get an alternative proof of
Gardiner's formula for the derivative of 
extremal length in some case.
Indeed,
this is the case where we need in the proof of Theorem \ref{thm:pluri-subharmonicity}.

Let $F\in \mathcal{MF}$ and $x_{0}=(M,f)\in \teich{g,m}$.
Suppose that the Hubbard-Masur differential $q_{0}=q_{F,x_{0}}$
is generic.
Let $v=[\dot{\mu}]$ and $\{q_{\lambda}\}_{\lambda\in D}$
as the proof of Theorem \ref{thm:leviform}.
Suppose $x_{\lambda}=\pi(q_{\lambda})$ varies holomorphically.
By applying the discussion in the proof of
Theorem \ref{thm:leviform},
from Lemma \ref{lem:transversal},
we have
\begin{align*}
&\left.
\frac{\partial}{\partial\lambda}
\ext_{x_{\lambda}}(F)
\right|_{\lambda=0}\\
&=
\frac{i}{4}
\sum_{k=1}^{\tilde{g}}
\left(
\int_{\alpha_{k}}\overline{
\left(
\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
\right)
}\int_{\beta_{k}}\overline{\omega_{q_{0}}}
-
\int_{\beta_{k}}\overline{
\left(
\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
\right)
}
\int_{\alpha_{k}}\overline{\omega_{q_{0}}}
\right) \\
&\quad
+
\frac{i}{4}
\sum_{k=1}^{\tilde{g}}
\left(
\int_{\alpha_{k}}\omega_{q_{0}}
\int_{\beta_{k}}
\overline{
\left(
-\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
\right)
}
-
\int_{\beta_{k}}\omega_{q_{0}}
\int_{\alpha_{k}}\overline{
\left(-
\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
\right)
}
\right) \\
&=
\frac{i}{4}
\int_{\tilde{M}_{q_{0}}}
\overline{
\left(
\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
\right)}
\wedge \overline{\omega_{q_{0}}}
-
\frac{i}{4}
\int_{\tilde{M}_{q_{0}}}\omega_{q_{0}}
\wedge 
\overline{
\left(
\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
\right)}
\\
&=-\frac{1}{2}
\int_{\tilde{M}_{q_{0}}}
\frac{\overline{(\pi_{q_{0}})^{*}(\eta_{v})}}{|\omega_{q_{0}}|^{2}}
\omega_{q_{0}}^{2}
%dxdy
%=\frac{1}{2}
%\int_{\tilde{M}_{q_{0}}}
%(\pi_{q_{0}})^{*}
%\left(
%\frac{\overline{\eta_{0}}}{|q_{0}|^{2}}
%q_{0}
%\right)
=
-\int_{M_{q_{0}}}
\frac{\overline{\eta_{v}}}{|q_{0}|}q_{0}
=
-\int_{M}
\dot{\mu}\,q_{F,x_{0}}
\end{align*}
because $M=M_{q_{0}}$ by definition.
%
%\subsection{Proof of Corollary \ref{coro:qualitative-disk-convexity}}
%\label{proof-of-disk-convexity}
%The quasifuchsian space $\mathcal{QF}$ is biholomorphically equivalent to
%the product of Teichm\"uller space 
%$\teich{g,m}\times\teich{g,m}$ (Bers' simultaneous uniformization theorem).
%Let $Q(x,\overline{y})$ be the quasifuchsian group uniformlized by $(x,\overline{y})\in \teich{g,m}\times\teich{g,m}$.
%The quasifuchsian group contains the fuchsian space
%$\mathcal{F}$ which consists of fuchsian groups
%as diagonal $\mathcal{F}=\{(z,\overline{z})\in \teich{g,m}\times\teich{g,m}\}$.
%Consider a function on $\mathcal{QF}$ defined by
%$$
%d_{(z,\overline{z})}(x,\overline{y})=\max\{d_{T}(z,x),d_{T}(\overline{z},\overline{y})\}.
%$$
%From Corollary \ref{coro:Teichmuller distance},
%$d_{(z,\overline{z})}$ is plurisubharmonic.
%Furthermore,
%$Q(x,\overline{y})$ is a $K$-quasifuchsian group
%if and only if there is $(z,\overline{z})\in \mathcal{F}$
%such that $d_{(z,\overline{z})}(x,\overline{y})\le
%(\log K)/2$.
%Set
%$$
%u(x,y)=\sup_{(z,\overline{z}\in \mathcal{F}}
%$$

%
%\subsection{Gemetric interpretation of $\eta_{v}$}
%\label{sec:geometric-interpretation}
%%
%%\subsubsection{Gemetric interpretation}
%%We chack that
%%%$D\ni \lambda\mapsto \chi_{q_{\lambda}}(c)$ is real analytic
%%%for every $c\in H_{1}(\tilde{M}_{q_{0}})$
%%$\eta_{\lambda}$ in Lemma \ref{lem:transversal} varies real analytically on $D$
%%when 
%Let $x_{0}=(M_{0},f_{0})\in \teich{g,m}$.
%Suppose $q_{0}=q_{F,x_{0}}$ is generic.
%%In \cite{GM},
%%Gardiner and Masur
%%observed that
%%the mapping
%%\begin{equation}
%%\label{eq:Gardiner-Masur}
%%\GM\colon
%%\quaddiff{g,m}\ni q\mapsto (\horizontal(q),\vertical(q))\in \mathcal{MF}\times \mathcal{MF}
%%\end{equation}
%%is injective,
%%where $\horizontal(q)$ is the horizontal foliation of $q$.
%%Namely,
%%$\horizontal(q)=\vertical(-q)$.
%%Let $\xi_{F,G}=\GM^{-1}(F,G)$
%%for $(F,G)\in \GM(\quaddiff{g,m})$.
%Let $v=[\dot{\mu}]\in T_{x_{0}}\teich{g,m}$.
%The quadratic differential $\eta_{v}\in \quaddiff{g,m}$
%is defined to satisfy
%\begin{equation}
%\label{eq:pairing-and-projection}
%\langle v,\psi\rangle=\int_{M_{0}}\frac{\overline{\eta_{v}}}{|q_{0}|}\psi
%\end{equation}
%for all $\psi\in \quaddiff{g,m}$
%(cf. Lemma \ref{lem:transversal} and Theorem \ref{thm:leviform}).
%The definition of $\eta_{v}$
%implies
%as if the infinitesimal Beltrami differential $\dot{\mu}$
%were infinitesimally equivalent to the ``infinitesimal deformation'' along
%$\overline{\eta_{v}}/|q_{0}|$
%on $M_{q_{0}}$
%(cf. \cite{MR1215481}).
%However,
%$\overline{\eta_{v}}/|q_{0}|$
%is not a Beltrami differential in general,
%becauce it may have $1/|z|$-singularities at zeroes of $q_{0}$
%and hence it could happen $\|\overline{\eta_{v}}/|q_{0}|\|_{\infty}=\infty$.
%Therefore,
%the above discussion does not interpret correctly
%the geometry of the map
%\begin{equation}
%\label{eq:corresponding-v-eta}
%T_{x_{0}}\teich{g,m}\ni v\mapsto \eta_{v}\in \quaddiff{g,m}
%\end{equation}
% in general.
%In this section,
%we shall try to give a geometric description of the correspondence
%\eqref{eq:corresponding-v-eta}.
%
%From \cite[Theorem 5.8]{2012arXiv1203.0273D} and \cite{GM},
%the map 
%\begin{equation}
%\label{eq:Gardiner-Masur}
%%\GM\colon
%\quaddiff{g,m}\ni q\mapsto (\horizontal(q),\vertical(q))\in \mathcal{MF}\times \mathcal{MF}
%\end{equation}
%is a real analytic diffeomorphism
%around $q_{0}$.
%%
%%Therefore,
%%we can obtain a real-analytic diffeomorphism
%Hence, the map
%\begin{equation}
%\label{eq:Gardiner-Masur2}
%\complexstructure\colon
%\teich{g,m}\times \teich{g,m}
%\ni (x,y)\mapsto -q_{\horizontal(q_{F,x}),y}\in 
%\quaddiff{g,m}
%\end{equation}
%is also a real analytic diffeomorphism on a neighborhood $N_{1}\times N_{1}$
%of $(x_{0},x_{0})$
%onto the image,
%where $N_{1}$ is a neighborhood of $x_{0}=\pi(q_{0})\in \teich{g,m}$.
%Notice that 
%$\horizontal(\complexstructure(x,y))=\horizontal(q_{F,x})$,
%$\complexstructure(x,y)\in \quaddiff{y}$
%and $\complexstructure(x,x)=q_{F,x}$ for all $x,y\in N_{1}$.
%Let $\complexstructure_{y}(x)=\complexstructure(x,y)$ for $(x,y)\in N_{1}\times N_{1}$
%(Figure \ref{fig:mapJ}).
%\begin{figure}
%\includegraphics[height=6cm]{mapJ.eps}
%\caption{A schematic of the map $\mathcal{J}_{x_{0}}$.}
%\label{fig:mapJ}
%\end{figure}
%
%
%%for $x,y\in N_{1}$,
%%the derivative of the map \eqref{eq:Gardiner-Masur2}
%%along the first coordinate gives a linear mapping
%%$$
%%L_{(x,y)}\colon T_{x}\teich{g,m}\to \quaddiff{y}
%%\cong
%%{\rm Ker}(\pi_{*}\mid_{\Xi(x,y)}).
%%$$
%
%\begin{theorem}[Geometric interpretation]
%\label{thm:duality}
%For $v\in T_{x_{0}}\teich{g,m}$,
%we have
%$$
%(\complexstructure_{x_{0}})_{*}[v]=-4\eta_{v},
%$$
%where $\eta_{v}\in \quaddiff{x_{0}}$ is taken
%as \eqref{eq:pairing-and-projection}.
%\end{theorem}
%
%\begin{proof}
%Consider the triangulation $\Delta_{q}$ and
%the train track $\tau_{q_{0}}$ on $M_{q}$
%for the horizontal foliation $|{\rm Im}(\sqrt{q})|$
%of $q$
%%as the proof of Lemma \ref{lem:constant-representation}
%(cf. \cite[Theorem 5.8]{2012arXiv1203.0273D}).
%We may assume that $\Delta_{q}$ contains no horizontal edge
%(cf. \cite[Lemma 5.7]{2012arXiv1203.0273D}).
%Consider the lifts $\tilde{\Delta}_{q}$ and $\tilde{\tau}_{q}$ of
%$\Delta_{q}$ and $\tau_{q}$ to $\tilde{M}_{q}$,
%respectivey.
%We orient each edge of $\tilde{\Delta}_{q}$ so that
%${\rm Im}(\omega_{q})$ is positive along the edge.
%Each branch $b$ of $\tilde{\tau}_{q}$ intersects
%a unique edge $e_{b}$ of $\tilde{\Delta}_{q}$.
%We orient $b$ so that $b\cdot e_{b}=+1$,
%where $\cdot$ means the algebraic intersection number.
%By taking $U$ sufficiently small,
%we may also assume that the marking $\tilde{M}_{q_{0}}\to \tilde{M}_{q}$
%induces isomorphisms from $\tilde{\Delta}_{q_{0}}$ and
%$\tilde{\tau}_{q_{0}}$
%to $\tilde{\Delta}_{q}$ and $\tilde{\tau}_{q}$,
%respectively.
%
%From \cite[Theorem 5.8]{2012arXiv1203.0273D},
%the tangential map of
%the map $\horizontal\colon \quaddiff{x_{0}}\to \mathcal{MF}$
%of the horizontal foliations
%satisfies 
%\begin{equation}
%\label{eq:Dumas1}
%%\frac{1}{2}
%%{\rm Im}\int_{\tilde{M}_{q_{0}}}
%%\frac{(\pi_{q_{0}})^{*}(\psi_{1})\overline{(\pi_{q_{0}})^{*}(\psi_{2})}}{4|\omega_{q_{0}}|^{2}}=
%{\rm Im}\int_{M_{q_{0}}}\frac{\psi_{1}\overline{\psi_{2}}}{4|q_{0}|}
%=\ThurstonSymplectic{\horizontal_{*}(\psi_{1})}{\horizontal_{*}(\psi_{2})}{\tau_{q_{0}}}
%\end{equation}
%for $\psi_{1},\psi_{2}\in 
%T_{q_{0}}\quaddiff{x_{0}}\cong \quaddiff{x_{0}}$,
%where $\ThurstonSymplectic{\cdot}{\cdot}{\tau_{q_{0}}}$
%is the Thurston's symplectic form on $W(\tau_{q_{0}})$,
%where $W(\tau_{q_{0}})$ is the space
%of real valued functions on the branches of $\tau_{q_{0}}$
%satisfying the switch condition.
%It is known that the space $W(\tau_{q_{0}})$
%is identified with the tangent space of $\mathcal{MF}$
%at $\horizontal(q_{0})$
%since $\tau_{q_{0}}$ is maximal
%(cf. \cite[\S3.2]{MR1144770}).
%%We can treat in the same way even if $\tilde{\tau'_{q_{0}}$ is a bigon track
%%(cf. \cite[\S3.4]{MR1144770}).
%
%From the definition,
%$|{\rm Im}(\omega_{q_{F,x}})|$ is the lift of the horizontal foliation
%of $q_{F,x}$ to $\tilde{M}_{q_{F,x}}$.
%By the definition of the orientation of edges of $\tilde{\Delta}_{q_{0}}$,
%the image of the map
%$N_{1}\ni x\mapsto \horizontal(q_{F,x})\in \mathcal{MF}$
%at $x$ is identified with $w_{x}\in W(\tilde{\tau}_{q_{0}})$
%which is defined by
%$$
%w_{x}(b)=\int_{b}|{\rm Im}(\omega_{q_{F,x}})|
%=\int_{b}{\rm Im}(\omega_{q_{F,x}}).
%$$
%From Lemmas \ref{lem:Douady-Hubbard} and \ref{lem:transversal},
%the image of $v\in T_{x_{0}}\teich{g,m}$ under the differential of the map
%$N_{1}\ni x\mapsto {\rm Im}(\chi_{q_{F,x}})$
%is identified with
%the homomorphism
%\begin{equation}
%\label{eq:derivative1}
%V_{v}
%\colon H_{1}(\tilde{M}_{q_{0}})
%\ni c\mapsto 
%-2{\rm Im}\int_{c}
%\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
%\in \mathbb{R}.
%\end{equation}
%Indeed,
%when $\{q_{\lambda}\}_{\lambda}\in D$
%is taken as Lemma \ref{lem:transversal},
%\eqref{eq:derivative1}
%is derived from
%\begin{align*}
%\frac{\partial {\rm Im}(\chi_{q_{\lambda}}(c))}{\partial t}
%&=
%-i\left(\frac{\partial \chi_{q_{\lambda}}(c)}{\partial \lambda}
%+\frac{\partial \chi_{q_{\lambda}}(c)}{\partial \overline{\lambda}}
%\right)
%\\
%&=-i\int_{c}
%\overline{
%\left(
%\frac{(\pi_{q_{\lambda}})^{*}(\eta_{\lambda})}{\omega_{q_{\lambda}}}
%\right)}
%-\frac{(\pi_{q_{\lambda}})^{*}(\eta_{\lambda})}{\omega_{q_{\lambda}}}
%\\
%&=-2{\rm Im}\int_{c}
%\frac{(\pi_{q_{\lambda}})^{*}(\eta_{\lambda})}{\omega_{q_{\lambda}}}
%\end{align*}
%where $\lambda=t+is$,
%since the real part of $\chi_{q_{\lambda}}$ is constant on $D$.
%
%Let $p\colon \tilde{\tau}_{q_{0}}\to \tau_{q_{0}}$ be the
%induced covering.
%For a weight $w\in W(\tau_{q_{0}})$,
%we define $C_{w}\in H_{1}(\tilde{M}_{q_{0}},\mathbb{R})^{-}$
%as the homology class of a cycle
%$$
%\sum_{\mbox{$b$:branches of $\tilde{\tau}_{q_{0}}$}}w(p(b))b.
%$$
%%Let $\sigma(c)=\sigma_{c}(z)dz+\overline{\sigma_{c}(z)}d\overline{z}$
%%be the reproducing harmonic differential on $\tilde{M}_{q_{0}}$
%%associated to $c\in H_{1}(\tilde{M}_{q_{0}})$,
%%where $\sigma_{c}(z)dz$ is a holomorphic $1$-form on $\tilde{M}_{q_{0}}$
%%(we call this holomorphic $1$-form the \emph{holomorphic part} of $\sigma(c)$).
%%Namely,
%%the differential $\sigma(c)$ is a unique harmonic differential on $\tilde{M}_{q_{0}}$
%%which satisfies
%%$$
%%\int_{c}\omega=\int_{\tilde{M}_{q_{0}}}\omega\wedge {}^{*}\overline{\sigma(c)}
%%$$
%%for all closed $1$-form $\omega$ on $\tilde{M}_{q_{0}}$.
%Then,
%\begin{align*}
%w_{x}(C_{w})
%&=
%\sum_{b}w(p(b))w_{x}(b)
%\\
%V_{v}(C_{w})
%&=
%-2{\rm Im}\int_{C_{w}}
%\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
%=
%-2
%\sum_{b}w(p(b))
%{\rm Im}\int_{b}
%\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}.
%%=
%%2{\rm Im}\int_{\tilde{M}_{q_{0}}}
%%\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}
%%\omega\wedge {}^{*}\overline{\sigma(C_{w})}
%\end{align*}
%These equations imply that
%the differential of the map $N_{1}\ni x\mapsto w_{x}\in W(\tilde{\tau}_{q_{0}})$ at $x=x_{0}$ is equal to the weight
%(which we abbreviate as)
%$$
%V_{v}(b)=-2\,{\rm Im}\int_{b}
%\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{\omega_{q_{0}}}.
%$$
%The weight $V_{v}$ is invariant under the action of covering
%transformation $r_{0}$ of $\tilde{M}_{q_{0}}\to M_{q_{0}}$
%on $\tilde{\tau}_{q_{0}}$,
%because
%for any branch $b$ of $\tilde{\tau}_{q_{0}}$,
%the orientation of $r_{0}(b)$ is opposite to the orientation induced
%from $b$.
%Thus,
%$V_{v}$ descends to a weight $p_{*}(V_{v})$
%on $\tau_{q_{0}}$.
%
%Let $\psi_{0}=(\complexstructure_{x_{0}})_{*}[v]$.
%For all $\psi\in \quaddiff{x_{0}}\subset T_{q_{0}}\quaddiff{g,m}$,
%we define $w(\psi)\in W(\tilde{\tau}_{q_{0}})$ by
%$$
%w(\psi)(b)={\rm Im}\int_{b}\frac{(\pi_{q_{0}})^{*}(\psi)}{2\omega_{q_{0}}}.
%$$
%Then,
%we have
%$\horizontal_{*}[\psi]=p_{*}(w(\psi))$
%(cf. \cite[Proof of Theorem 5.8]{2012arXiv1203.0273D}).
%By the definition of $\complexstructure_{x_{0}}$,
%$\horizontal\circ \complexstructure_{x_{0}}(x)=\horizontal(q_{F,x})$
%for all $x\in N_{1}$.
%Hence,
%$\horizontal_{*}[\psi_{0}]=p_{*}(V_{v})$
%as elements of $W(\tilde{\tau}_{q_{0}})
%\cong T_{\horizontal(q_{0})}\mathcal{MF}$.
%From \eqref{eq:Dumas1}
%and \cite[(5.4)]{2012arXiv1203.0273D},
%the following holds for all $\psi\in \quaddiff{x_{0}}$:
%\begin{align*}
%{\rm Im}\int_{M_{q_{0}}}\frac{\psi_{0}\overline{\psi}}{4|q_{0}|}
%&=\frac{1}{2}\,{\rm Im}\int_{\tilde{M}_{q_{0}}}
%\frac{(\pi_{q_{0}})^{*}(\psi_{0})\overline{(\pi_{q_{0}})^{*}(\psi)}}
%{4|\omega_{q_{0}}|^{2}}
%\\
%&=
%\frac{1}{2}\,\ThurstonSymplectic{V_{v}}{w(\psi)}{\tilde{\tau}_{q_{0}}} \\
%&=
%-2\int_{\tilde{M}_{q_{0}}}
%{\rm Im}\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{2\omega_{q_{0}}}
%\wedge
%{\rm Im}\frac{(\pi_{q_{0}})^{*}(\psi)}{2\omega_{q_{0}}} \\
%&=
%-{\rm Re}\int_{\tilde{M}_{q_{0}}}
%\frac{(\pi_{q_{0}})^{*}(\eta_{v})}{2\omega_{q_{0}}}
%\wedge
%\overline{\frac{(\pi_{q_{0}})^{*}(\psi)}{2\omega_{q_{0}}}}
%\\
%&=
%-2{\rm Im}\int_{\tilde{M}_{q_{0}}}
%\frac{(\pi_{q_{0}})^{*}(\eta_{v})\overline{(\pi_{q_{0}})^{*}(\psi)}}
%{4|\omega_{q_{0}}|^{2}}
%=
%-{\rm Im}\int_{M_{q_{0}}}\frac{\eta_{v}\overline{\psi}}{|q_{0}|}.
%\end{align*}
%Therefore,
%we have
%$(\complexstructure_{x_{0}})_{*}[v]
%=\psi_{0}=-4\eta_{v}$.
%%%which is defined by
%%%$$
%%%%\frac{\partial \chi_{q_{\lambda}}(c)}{\partial t}
%%%%=
%%%%\frac{\partial \chi_{q_{\lambda}}(c)}{\partial \lambda}
%%%%+
%%%%\frac{\partial \chi_{q_{\lambda}}(c)}{\partial \overline{\lambda}}
%%%V_{v}(c)=
%%%2i{\rm Im}\int_{c}
%%%\frac{(\pi_{q_{0}})^{*}(\psi_{1})}{\omega_{q_{0}}}
%%%$$
%%for $c\in H_{1}(\tilde{M}_{q_{0}})$,
%%where $\lambda=t+is$.
%\end{proof}

\subsection{Examples}
\label{subsec:examples}
In this section,
we treat the exceptional cases
in the proofs of pluriharmonicity in the previous sections.
\subsubsection{Levi forms for the case of flat tori}
\label{subsubsec:tori}
We start to consider
the Levi forms of extremal length functions on the Teichm\"uller space
of flat tori (once punctured tori),
and check our formula in Theorem \ref{thm:leviform} also valid in this case.
This case is excluded from our assumption in the theorem.
However,
this is a motivated example of our result.

The Teichm\"uller space of a flat torus is identified with the upper half plane $\mathbb{H}$.
A point $\tau\in \mathbb{H}$ corresponds to a marked Riemann surface
$M_{\tau}$
which is defined as the quotient space of $\mathbb{C}$
by a lattice generated by $z+1$ and $z+\tau$.
%Let $\tau\in \mathbb{H}$.
Notice that the space $\mathcal{PMF}=\mathcal{PMF}(M_{i})$
of projective measured foliations on $M_{i}$
($i=\sqrt{-1}\in \mathbb{H}$)
is identified with the projective space $\mathbb{RP}^{1}=\{[x\colon y]\mid
(x,y)\ne (0,0)\}$.
A point $[x\colon y]\in \mathbb{RP}^{1}$
corresponds to
a measured foliation whose leaves are parallel to the line of direction $x+yi$.
%For instance,
%the above curve $\beta$ corresponds to $[0\colon 1]$.

Let $\alpha$ be a closed curve on $M_{i}$
corresponding to $[1\colon 0]\in \mathcal{PMF}$.
%associated to a horizontal line,
%where
%$i=\sqrt{-1}\in \mathbb{H}$.
Then,
the Jenkins-Strebel differential $q_{\alpha,\tau}$ on $M_{\tau}$
for $\alpha$
and
the extremal length function of $\alpha$ at $M_{\tau}$
are obtained as
\begin{align*}
q_{\alpha,\tau}&=-\frac{1}{{\rm Im}(\tau)^{2}}dz^{2}\\
\ext_{\tau}(\alpha)
&=\frac{1}{{\rm Im}(\tau)}.
\end{align*}
Fix $\tau_{0}\in \mathbb{H}$.
For $V\in \mathbb{C}$,
let $v$ be the tangent vector at $\tau_{0}$
corresponding to the infinitesimal quasiconformal
deformation
from $M_{\tau_{0}}$ to $M_{\tau_{0}+\lambda V}$ as $\lambda\to 0$.
The Levi form (Laplacian) of the extremal length function is
\begin{equation}
\label{eq:laplacian-extremal-length-function1}
\levi{\ext_{\cdot}(\alpha)}[v,\overline{v}]=
\left.
\frac{\partial^{2}}{\partial \lambda\partial\overline{\lambda}}
\ext_{\tau_{0}+\lambda V}(\lambda)
\right|_{\lambda=0}
=\frac{|V|^{2}}{2{\rm Im}(\tau_{0})^{3}}
\end{equation}
The Beltrami differential $\mu(\lambda)$
of the quasiconformal deformation
from $M_{\tau_{0}}$ to $M_{\tau_{0}+\lambda V}$
behaves
$$
\mu(\lambda)=\lambda\cdot \dot{\mu}+o(|\lambda|)=
\lambda\cdot \frac{iV}{2{\rm Im}(\tau_{0})}\frac{d\overline{z}}{dz}+o(|\lambda|)
$$
as $t\to 0$.
By definition,
$v=[\dot{\mu}]$.
We define
a holomorphic quadratic differential $\eta_{v}$ on $M_{\tau_{0}}$ by 
\begin{equation}
\label{eq:laplacian-extremal-length-function3}
\eta_{v}=-\frac{i\overline{V}}{2{\rm Im}(\tau_{0})^{3}}dz^{2}.
\end{equation}
This differential satisfies that
$$
\langle v,\psi\rangle=
\int_{M_{\tau_{0}}}\dot{\mu}\psi=
\int_{M_{\tau_{0}}}\frac{\overline{\eta_{v}}}{|q_{\alpha,\tau_{0}}|}\psi
$$
for all holomorphic quadratic differential $\psi$ on $M_{\tau_{0}}$.
%where
%$$
%q_{F,\tau}=\frac{e^{i\theta(\tau)}}{{\rm Im}(\tau)^{2}}dz^{2}
%$$
%is the Hubbard-Masur differential on $M_{\tau}$
%associated to $F$
%($\theta(\tau)\in \mathbb{R}$).
Therefore,
the right-hand side of the formula in Theorem \ref{thm:leviform}
is equal to
$$
2\int_{M_{\tau_{0}}}\frac{|\eta_{v}|^{2}}{|q_{\alpha,\tau_{0}}|}
=\frac{|V|^{2}}{2{\rm Im}(\tau_{0})^{3}},
$$
which coincides with the right-hand side of \eqref{eq:laplacian-extremal-length-function1}.

Let $F$ be a measured foliation
corresponding to $[a\colon b]\in \mathcal{PMF}$ with $b\ne 0$.
We normalize $F$ with $i(\alpha,F)=1$.
Then the Hubbard-Masur differential $q_{F,\tau}$ on $M_{\tau}$
and the extremal length function for $F$
are
\begin{align*}
q_{F,\tau}
&=-\frac{(a+b\overline{\tau})^{2}}{b^{2}{\rm Im}(\tau)^{2}}dz^{2}
\\
\ext_{\tau}(F)
&=
\frac{|a+b\tau|^{2}}{b^{2}{\rm Im}(\tau)}.
\end{align*}
Let $v$ be the tangent vector at $\tau_{0}$
corresponding to
$\dot{\mu}=\frac{iV}{2{\rm Im}(\tau_{0})}\frac{d\overline{z}}{dz}$.
Then,
\begin{equation}
\label{eq:laplacian-extremal-length-function2}
\levi{\ext_{\cdot}(\alpha)}[v,\overline{v}]=
\left.
\frac{\partial^{2}}{\partial \lambda\partial\overline{\lambda}}
\ext_{\tau_{0}+\lambda V}(\lambda)
\right|_{\lambda=0}
=\frac{|a+b\tau_{0}|^{2}}{2b^{2}{\rm Im}(\tau_{0})^{3}}|V|^{2}
\end{equation}
In this case,
the differential $\eta_{v}\in \quaddiff{\tau_{0}}$
is
\begin{equation}
\label{eq:laplacian-extremal-length-function4}
\eta_{v}=-\frac{i|a+b\tau_{0}|^{2}
\overline{V}}{2b^{2}{\rm Im}(\tau_{0})^{3}}dz^{2}.
\end{equation}
and hence
$$
2\int_{M_{\tau_{0}}}\frac{|\eta_{v}|^{2}}{|q_{F,\tau_{0}}|}
=\frac{|a+b\tau_{0}|^{2}}{2b^{2}{\rm Im}(\tau_{0})^{3}}|V|^{2},
$$
which coincides with the right-hand side of \eqref{eq:laplacian-extremal-length-function2}.

\subsubsection{The gemetric interpretation of $\eta_{v}$ for the case of tori}
\label{subsubsec:example-tori}
We check Theorem \ref{thm:duality} in the case of flat tori
(once punctured tori).
We use the notation defined in \S\ref{subsubsec:tori} frequently.
%We identify the Teichm\"uller space with the upper half plane
%$\mathbb{H}$.
%We define a torus $M_{\tau}$ for $\tau\in \mathbb{H}$
%as \S\ref{subsubsec:tori}.
%Let $\beta$ be a simple closed curve on $M_{\sqrt{-1}}$
%associated to a vertical line.
%Notice that the space of projective measured foliations on $M_{\sqrt{-1}}$
%is identified with the projective space $\mathbb{RP}^{1}=\{[x\colon y]\mid
%(x,y)\ne (0,0)\}$.
%A point $[x\colon y]\in \mathbb{RP}^{1}$
%corresponds to
%a measured foliation whose leave are parallel to the line of direction $x+yi$.
%For instance,
%the above curve $\beta$ corresponds to $[0\colon 1]$.

First we consider the case $\alpha=[1\colon 0]$.
%The Jenkins-Strebel differential $q_{\beta,\tau}$ on
%$M_{\tau}$ for $\beta$
%is equal to
%$$
%q_{\beta,\tau}=-\frac{\overline{\tau}^{2}}{{\rm Im}(\tau)^{2}}dz^{2},
%$$
%and the projective class of the
Fix $\tau_{0}\in \mathbb{H}$.
Since
the projective class of the horizontal foliation of $q_{\alpha,\tau}$ corresponds to
a projective class $[-{\rm Re}(\tau)\colon 1]\in \mathbb{RP}^{1}$,
the underlying foliation of
the horizontal foliation $J_{\tau_{0}}(\tau)$
is foliated by
the lines (leaves) directed to $-({\rm Re}(\tau))+\tau_{0}$.
Since the horizontal foliations of $J_{\tau_{0}}(\tau)$ and $q_{\alpha,\tau}$
are same,
$$
\int_{\alpha}|{\rm Im}\sqrt{J_{\tau_{0}}(\tau)}|
=i(\horizontal(J_{\tau_{0}}(\tau)),\alpha)
=i(\horizontal(q_{\alpha,\tau}),\alpha)
=
\int_{\alpha}|{\rm Im}\sqrt{q_{\alpha,\tau}}|
$$
holds.
Therefore,
we can check that
$$
J_{\tau_{0}}(\tau)=
\left(
\frac{-{\rm Re}(\tau)+\overline{\tau_{0}}}
{{\rm Im}(\tau){\rm Im}(\tau_{0})}
\right)^{2}
dz^{2}
\in \quaddiff{\tau_{0}}
$$
for $\tau\in \mathbb{H}$.
Let $v=[\dot{\mu}]$ be a tangent vector at $\tau_{0}$,
where
$\dot{\mu}=\frac{iV}{2{\rm Im}(\tau_{0})}\frac{d\overline{z}}{dz}$
with $V\in \mathbb{C}$
as \S\ref{subsubsec:tori}.
%Therefore,
%for $v\in T_{\tau_{0}}\mathbb{H}$,
Then,
from the above calculation,
we obtain
$$
(J_{\tau_{0}})_{*}[v]=
\left(2\frac{i\overline{V}}{{\rm Im}(\tau_{0})^{3}}
\right)dz^{2}
\in \quaddiff{\tau_{0}}.
$$
Comparing with \eqref{eq:laplacian-extremal-length-function3},
we get $(J_{\tau_{0}})_{*}[v]=-4\eta_{v}$
as we appeared in Theorem \ref{thm:duality}.

Let $F\in \mathcal{PMF}$ be a measured foliation
corresponding to $[a\colon b]\in \mathcal{PMF}$ with $b\ne 0$
and $i(\alpha,F)=1$.
Then,
the underlying foliation of the horizontal foliation of $q_{F,\tau}$
is foliated by the lines of direction $[a{\rm Re}(\tau)+b|\tau|^{2}\colon a+b{\rm Re}(\tau)]$.
By the similar argument as above,
one can check that
\begin{align*}
J_{\tau_{0}}(\tau)
&=\left(
\frac{-(a{\rm Re}(\tau)+b|\tau|^{2})+(a+b{\rm Re}(\tau))\overline{\tau_{0}}}
{b{\rm Im}(\tau){\rm Im}(\tau_{0})}
\right)^{2}
dz^{2}
\\
(J_{\tau_{0}})_{*}[v]
&=
\left(2\frac{i|a+b\tau_{0}|^{2}\overline{V}}{b^{2}{\rm Im}(\tau_{0})^{3}}
\right)dz^{2}
\in \quaddiff{\tau_{0}}.
\end{align*}
%
%By definition
%$$
%\int_{\alpha}|{\rm Im}\sqrt{J_{\tau_{0}}(\tau)}|=
%\int_{\alpha}|{\rm Im}\sqrt{q_{F,\tau}}|,
%$$
%and hence
%take $\eta_{v}\in \quaddiff{\tau_{0}}$
%to satisfy
%$$
%\langle v,\varphi\rangle
%=\int_{M_{\tau_{0}}}\dot{\mu}\varphi
%=
%\int_{M_{\tau_{0}}}\frac{\overline{\eta_{v}}}{|q_{\beta,\tau_{0}}|}\varphi
%$$
%for all $\varphi\in \quaddiff{\tau_{0}}$.
%%where
%%$\dot{\mu}=\frac{i\alpha}{2{\rm Im}(\tau_{0})}\frac{d\overline{z}}{dz}$
%One can check that
%$$
%\eta_{v}
%=-\frac{i|\tau_{0}|^{2}\overline{V}}{2{\rm Im}(\tau_{0})^{3}}dz^{2}
%$$
From \eqref{eq:laplacian-extremal-length-function4},
we also have
$(J_{\tau_{0}})_{*}[v]=-4\eta_{v}$.
%as we appeared in Theorem \ref{thm:duality}.
%
%For a measured foliation $F=[a\colon b]$ with $b\ne 0$,
%one can check that
%$$
%J_{\tau_{0}}(\tau)=
%$$


%
%Fix $\tau_{0}\in \mathbb{H}$.
%From the discussion above,
%the underlying foliation of
%the horizontal foliation $J_{\tau_{0}}(\tau)$
%is foliated by
%the lines (leaves) directed to $-|\tau|^{2}+({\rm Re}(\tau))\tau_{0}$.
%Since the horizontal foliations of $J_{\tau_{0}}(\tau)$ and $q_{\beta,\tau}$
%are same,
%$$
%\int_{\beta}|{\rm Im}\sqrt{J_{\tau_{0}}(\tau)}|
%=
%\int_{\beta}|{\rm Im}\sqrt{q_{\beta,\tau}}|
%$$
%holds.
%Therefore,
%we can check that
%$$
%J_{\tau_{0}}(\tau)=
%\left(\frac{|\tau|^{2}-({\rm Re}(\tau))\overline{\tau_{0}}}
%{{\rm Im}(\tau){\rm Im}(\tau_{0})}\right)^{2}
%dz^{2}
%\in \quaddiff{\tau_{0}}
%$$
%for $\tau\in \mathbb{H}$.
%Let $v=[\dot{\mu}]$ be a tangent vector at $\tau_{0}$,
%where
%$\dot{\mu}=\frac{iV}{2{\rm Im}(\tau_{0})}\frac{d\overline{z}}{dz}$
%with $V\in \mathbb{C}$
%as \S\ref{subsubsec:tori}.
%%Therefore,
%%for $v\in T_{\tau_{0}}\mathbb{H}$,
%Then,
%from the above calculation,
%we obtain
%$$
%(J_{\tau_{0}})_{*}[v]=
%\left(2\frac{i|\tau_{0}|^{2}\overline{V}}{{\rm Im}(\tau_{0})^{3}}
%\right)dz^{2}
%\in \quaddiff{\tau_{0}}.
%$$
%On the other hand,
%take $\eta_{v}\in \quaddiff{\tau_{0}}$
%to satisfy
%$$
%\langle v,\varphi\rangle
%=\int_{M_{\tau_{0}}}\dot{\mu}\varphi
%=
%\int_{M_{\tau_{0}}}\frac{\overline{\eta_{v}}}{|q_{\beta,\tau_{0}}|}\varphi
%$$
%for all $\varphi\in \quaddiff{\tau_{0}}$.
%%where
%%$\dot{\mu}=\frac{i\alpha}{2{\rm Im}(\tau_{0})}\frac{d\overline{z}}{dz}$
%One can check that
%$$
%\eta_{v}
%=-\frac{i|\tau_{0}|^{2}\overline{V}}{2{\rm Im}(\tau_{0})^{3}}dz^{2}
%$$
%and hence $(J_{\tau_{0}})_{*}[v]=-4\eta_{v}$
%holds as we appeared in Theorem \ref{thm:duality}.


\subsubsection{A comment on the case of $3g-3+m=1$}
The case of once punctured tori is treated in the same way
as that in the case of flat tori.
Indeed,
the canonical completion at the puncture gives an isomorphism between
the Teichm\"uller space of once punctured tori and that of flat tori.
The extremal length functions in the both spaces
are identified under the isomorphism.
The case of fours punctured sphere
is treated in the similar way.
%let us uniformize a fourth punctured sphere
%as a pillow $\mathbb{C}-\Gamma_{\tau}(\{0,\tau\})/\Gamma_{\tau}$
%with $\tau\in \mathbb{H}$,
%where $\Gamma_{\tau}$ is a subgroup of rigid motions of $\mathbb{C}$
%$$
%\Gamma_{\tau}=\langle z+1,z+2\tau,-z+2\tau\rangle.
%$$
%Under the uniformization,
%the Teichm\"uller space of fourth punctured spheres
%is identified with $\mathbb{H}$,
%and the extremal length function forms
%$$
%\frac{C}{{\rm Im}(\tau)}
%$$
%with some $C>0$.
%This means that the same calculation in the previous section
%also works in ths case.



%%%%%%%%%  Reference %%%%%%%%

\bibliographystyle{plain}
\bibliography{References-2}
%



\end{document}
Extremal_length_function.bbl
\def\cprime{$'$} \def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{MR534932}
Gunnar Berg.
\newblock Hyperconvexity and the {C}arath\'eodory metric.
\newblock {\em Arch. Math. (Basel)}, 32(2):189--191, 1979.

\bibitem{MR0168800}
Lipman Bers and Leon Ehrenpreis.
\newblock Holomorphic convexity of {T}eichm\"uller spaces.
\newblock {\em Bull. Amer. Math. Soc.}, 70:761--764, 1964.

\bibitem{MR847953}
Francis Bonahon.
\newblock Bouts des vari\'et\'es hyperboliques de dimension {$3$}.
\newblock {\em Ann. of Math. (2)}, 124(1):71--158, 1986.

\bibitem{MR0396936}
A.~Douady and J.~Hubbard.
\newblock On the density of {S}trebel differentials.
\newblock {\em Invent. Math.}, 30(2):175--179, 1975.

\bibitem{MR568308}
Fathi A. Fried F. Laudenbach F. Poénaru~V. Douady, A. and M.~Shub.
\newblock {\em Travaux de {T}hurston sur les surfaces}, volume~66 of {\em
  Ast\'erisque}.
\newblock S{\'e}minaire Orsay.

\bibitem{2012arXiv1203.0273D}
D.~{Dumas}.
\newblock {Skinning maps are finite-to-one}.
\newblock {\em ArXiv e-prints}, March 2012.

\bibitem{MR0352450}
Clifford~J. Earle.
\newblock On the {C}arath\'eodory metric in {T}eichm\"uller spaces.
\newblock In {\em Discontinuous groups and {R}iemann surfaces ({P}roc. {C}onf.,
  {U}niv. {M}aryland, {C}ollege {P}ark, {M}d., 1973)}, pages 99--103. Ann. of
  Math. Studies, No. 79. Princeton Univ. Press, Princeton, N.J., 1974.

\bibitem{MR1139765}
H.~M. Farkas and I.~Kra.
\newblock {\em Riemann surfaces}, volume~71 of {\em Graduate Texts in
  Mathematics}.
\newblock Springer-Verlag, New York, second edition, 1992.

\bibitem{MR903027}
Frederick~P. Gardiner.
\newblock {\em Teichm\"uller theory and quadratic differentials}.
\newblock Pure and Applied Mathematics (New York). John Wiley \& Sons, Inc.,
  New York, 1987.
\newblock A Wiley-Interscience Publication.

\bibitem{MR1099913}
Frederick~P. Gardiner and Howard Masur.
\newblock Extremal length geometry of {T}eichm\"uller space.
\newblock {\em Complex Variables Theory Appl.}, 16(2-3):209--237, 1991.

\bibitem{MR0102797}
Roger Godement.
\newblock {\em Topologie alg\'ebrique et th\'eorie des faisceaux}.
\newblock Actualit'es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13.
  Hermann, Paris, 1958.

\bibitem{MR507725}
Phillip Griffiths and Joseph Harris.
\newblock {\em Principles of algebraic geometry}.
\newblock Wiley-Interscience [John Wiley \& Sons], New York, 1978.
\newblock Pure and Applied Mathematics.

\bibitem{MR523212}
John Hubbard and Howard Masur.
\newblock Quadratic differentials and foliations.
\newblock {\em Acta Math.}, 142(3-4):221--274, 1979.

\bibitem{MR1215481}
Y.~Imayoshi and M.~Taniguchi.
\newblock {\em An introduction to {T}eichm\"uller spaces}.
\newblock Springer-Verlag, Tokyo, 1992.

\bibitem{MR499279}
Yujiro Kawamata.
\newblock On deformations of compactifiable complex manifolds.
\newblock {\em Math. Ann.}, 235(3):247--265, 1978.

\bibitem{MR559474}
Steven~P. Kerckhoff.
\newblock The asymptotic geometry of {T}eichm\"uller space.
\newblock {\em Topology}, 19(1):23--41, 1980.

\bibitem{MR1150978}
Maciej Klimek.
\newblock {\em Pluripotential theory}, volume~6 of {\em London Mathematical
  Society Monographs. New Series}.
\newblock The Clarendon Press, Oxford University Press, New York, 1991.
\newblock Oxford Science Publications.

\bibitem{MR815922}
Kunihiko Kodaira.
\newblock {\em Complex manifolds and deformation of complex structures}, volume
  283 of {\em Grundlehren der Mathematischen Wissenschaften [Fundamental
  Principles of Mathematical Sciences]}.
\newblock Springer-Verlag, New York, 1986.
\newblock Translated from the Japanese by Kazuo Akao, With an appendix by
  Daisuke Fujiwara.

\bibitem{MR1119946}
S.~L. Krushkal{\cprime}.
\newblock Strengthening pseudoconvexity of finite-dimensional {T}eichm\"uller
  spaces.
\newblock {\em Math. Ann.}, 290(4):681--687, 1991.

\bibitem{MR1142683}
Samuel~L. Krushkal.
\newblock The {G}reen function of {T}eichm\"uller spaces with applications.
\newblock {\em Bull. Amer. Math. Soc. (N.S.)}, 27(1):143--147, 1992.

\bibitem{MR0243112}
P.~Lelong.
\newblock {\em Fonctions plurisousharmoniques et formes diff\'erentielles
  positives}.
\newblock Gordon \& Breach, Paris-London-New York (Distributed by Dunod
  \'editeur, Paris), 1968.

\bibitem{MR2838267}
Anna Lenzhen and Kasra Rafi.
\newblock Length of a curve is quasi-convex along a {T}eichm\"uller geodesic.
\newblock {\em J. Differential Geom.}, 88(2):267--295, 2011.

\bibitem{2012arXiv1210.0743L}
L.~{Liu} and W.~{Su}.
\newblock {Variations of extremal length function on Teichmuller space}.
\newblock {\em ArXiv e-prints}, October 2012.

\bibitem{MR1283559}
Howard Masur.
\newblock The {T}eichm\"uller flow is {H}amiltonian.
\newblock {\em Proc. Amer. Math. Soc.}, 123(12):3739--3747, 1995.

\bibitem{MR1135877}
Howard Masur and John Smillie.
\newblock Hausdorff dimension of sets of nonergodic measured foliations.
\newblock {\em Ann. of Math. (2)}, 134(3):455--543, 1991.

\bibitem{MR1478844}
Curtis~T. McMullen.
\newblock Complex earthquakes and {T}eichm\"uller theory.
\newblock {\em J. Amer. Math. Soc.}, 11(2):283--320, 1998.

\bibitem{MR1231968}
Yair~N. Minsky.
\newblock Teichm\"uller geodesics and ends of hyperbolic {$3$}-manifolds.
\newblock {\em Topology}, 32(3):625--647, 1993.

\bibitem{MR1210002}
Takeo Ohsawa.
\newblock On the {B}ergman kernel of hyperconvex domains.
\newblock {\em Nagoya Math. J.}, 129:43--52, 1993.

\bibitem{MR1144770}
R.~C. Penner and J.~L. Harer.
\newblock {\em Combinatorics of train tracks}, volume 125 of {\em Annals of
  Mathematics Studies}.
\newblock Princeton University Press, Princeton, NJ, 1992.

\bibitem{MR0344979}
Tibor Rad{\'o}.
\newblock {\em On the problem of {P}lateau. {S}ubharmonic functions}.
\newblock Springer-Verlag, New York-Heidelberg, 1971.
\newblock Reprint.

\bibitem{MR766636}
Hiroshige Shiga.
\newblock On analytic and geometric properties of {T}eichm\"uller spaces.
\newblock {\em J. Math. Kyoto Univ.}, 24(3):441--452, 1984.

\bibitem{MR0374493}
Jean-Luc Stehl{\'e}.
\newblock Fonctions plurisousharmoniques et convexit\'e holomorphe de certains
  fibr\'es analytiques.
\newblock {\em C. R. Acad. Sci. Paris S\'er. A}, 279:235--238, 1974.

\bibitem{Thuston-LectureNote}
William {Thurston}.
\newblock {\em The {G}eometry and {T}opology of {T}hree-{M}anifolds}.
\newblock {L}ecture {N}ote at {P}rinceton {U}niversity, 1980. {A}vailable at
  {\tt http://library.msri.org/nonmsri/gt3m/}.

\bibitem{MR866707}
William~A. Veech.
\newblock The {T}eichm\"uller geodesic flow.
\newblock {\em Ann. of Math. (2)}, 124(3):441--530, 1986.

\bibitem{MR1094714}
William~A. Veech.
\newblock Moduli spaces of quadratic differentials.
\newblock {\em J. Analyse Math.}, 55:117--171, 1990.

\bibitem{MR880186}
Scott~A. Wolpert.
\newblock Geodesic length functions and the {N}ielsen problem.
\newblock {\em J. Differential Geom.}, 25(2):275--296, 1987.

\end{thebibliography}
