A note on mean curvature, Maslov class and symplectic area of Lagrangian immersions
In this note we prove a simple relation between the mean curvature form, symplectic area, and the Maslov class of a Lagrangian immersion in a K\"ahler-Einstein manifold. An immediate consequence is that in K\"ahler-Einstein manifolds with positive scalar curvature, minimal Lagrangian immersions are monotone.
math0310046.tex
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\begin{document}
\renewcommand{\abstractname}{Abstract}
\begin{abstract}
In this note we prove a simple relation between the mean curvature
form, symplectic area, and the Maslov class of a Lagrangian immersion
in a K\"ahler-Einstein manifold. An immediate consequence is that in
K\"ahler-Einstein manifolds with positive scalar curvature, minimal
Lagrangian immersions are monotone.  
\end{abstract}
\title[A note on mean curvature ...]{A note on mean curvature, Maslov
  class and symplectic area  of Lagrangian immersions}     
\authors{Kai Cieliebak\footnote{kai@mathematik.uni-muenchen.de} and Edward Goldstein\footnote{egold@math.stanford.edu}}
\maketitle


%%%
\section{Introduction}
Let $(M,\omega)$ be a K\"ahler-Einstein manifold whose Ricci curvature
is a multiple of the metric by a real number $\lambda$. Hence the
K\"ahler form $\omega$ and the first Chern class $c_1(M)$ are related
by $c_1(M)=\lambda[\omega]$. Let $L$ be an immersed Lagrangian
submanifold of $M$ 
and let $\sigma_L$ be the mean curvature form of $L$ (which is a
closed 1-form on $L$). Let $F:\Sigma\to M$ be a smooth map from
a compact connected surface to $M$ whose boundary $\partial F$ is contained in
$L$. Let $\mu(F)$ be the Maslov class of $F$ and $\omega(F)$ its
symplectic area. The goal of this note is to prove the following
simple relation between these quantities:
\begin{equation}\label{eq:main}
   \mu(F) - 2\lambda\omega(F)=\frac{\sigma_L(\partial F)}{\pi}.
\end{equation}

This relation was given in~\cite{M} for $\mathbb{C}^n$ and in~\cite{Ar}
for Calabi-Yau manifolds. Dazord \cite{D} showed
that the differential of the mean curvature form is the Ricci form, so
in the K\"ahler-Einstein case $\sigma_L$ is closed. Y.G. Oh~\cite{Oh2}
investigated the symplectic area in the case that the mean curvature
form is exact.  

In the case $\lambda>0$, Lagrangian submanifolds for which the
left-hand side vanishes on all disks $F$ are called {\em monotone} in
the symplectic geometry literature, cf.~\cite{Oh1}. An immediate
consequence of (\ref{eq:main}) is that in K\"ahler-Einstein manifolds
with positive scalar curvature, minimal Lagrangian immersions are
monotone.  

In view of the condition $c_1(M)=\lambda[\omega]$, the left-hand side
of (\ref{eq:main}) depends only on the boundary of $F$. Thus if the map
$H_1(L;\mathbb{R})\to H_1(M;\mathbb{R})$ is trivial it defines a
cohomology class 
$\delta_L\in H^1(L;\mathbb{R})$ via $\delta_L(\gamma) := \mu(F) -
2\lambda\omega(F)$ for some 2-cycle $F$ with $\partial F=\gamma$. It follows
that in this case the cohomology class of the mean curvature form
$\sigma_L$ is invariant under symplectomorphisms of $M$. This
generalizes Oh's observation~\cite{Oh2} 
that the cohomology class is invariant under Hamiltonian
deformations. One consequence is the following:

Let $(M,\omega)$ be a K\"ahler manifold with $c_1(M)=\lambda
[\omega]\in H_2(M;\mathbb{R})$. Let $L$ be an immersed Lagrangian
submanifold of $M$ such that the map $H_1(L;\mathbb{R})\to
H_1(M;\mathbb{R})$ is trivial and $\delta_L\neq 0$. Suppose
there is a K\"ahler-Einstein metric $\omega_{KE}$ in the same
cohomology class as $\omega$ and $\phi:(M,\omega)
\rightarrow (M,\omega_{KE})$ is a symplectomorphism (e.g.~the one
provided by Moser's theorem). Then $\phi(L)$ is Lagrangian but {\em
not} minimal.  
 
Note that for $\lambda \neq 0$ most Lagrangian submanifolds $L$ with
nontrivial first Betti number such that  
$H_1(L)\to H_1(M)$ vanishes have $\delta_L\neq 0$: For any
such $L$, pick a normal vector field $v$ to $L$ such that $i_v
\omega$ is closed on $L$ and non-trivial cohomologically. Then small
time variations of $L$ through $v$ produce Lagrangian submanifolds
with nontrivial $\delta_L$.


\section{Notation}
We first recall the definition of the Maslov index that is suitable
for our purposes. Let $V$ be a Hermitian vector space of complex
dimension $n$. Let $\Lambda^{(n,0)}V$ be the (one-dimensional)
space of holomorphic $(n,0)$-forms on $V$ and set
$$
K^2(V):=\Lambda^{(n,0)}V\otimes\Lambda^{(n,0)}V.
$$
Let $L$ be a Lagrangian subspace of $V$. We can associate to $L$ an
element $\kappa(L)$ in $\Lambda^{(n,0)}V$ of unit length which
restricts to a real volume form on $L$. This element is unique up to
sign and therefore defines a unique element of unit length
$$
   \kappa^2(L):=\kappa(L)\otimes\kappa(L)\in K^2(V).
$$
Thus we get a map $\kappa^2$ from the Grassmanian $Gr_{\rm Lag}(V)$ of
Lagrangian planes to the unit circle in $K^2(V)$. This map induces
a homomorphism $\kappa^2_{\ast}$ of fundamental groups 
\[
   \kappa^2_{\ast}:\pi_1(Gr_{\rm Lag}(V)) \rightarrow \mathbb{Z}.
\] 
To understand the map $\kappa^2_{\ast}$, let $L$ be a Lagrangian subspace
and let $v_1,\ldots,v_n$ be an orthonormal basis for $L$. For $0 \leq
t \leq 1$ consider the subspace
\[
   L_t=span\{v_1,\ldots,v_{n-1},e^{\pi i t}v_n\}.
\] 
This loop $\{L_t\}$ is the standard generator of $\pi_1 (Gr_{\rm
  Lag}(V))$. The induced elements in $\Lambda^{(n,0)}V$ are related by
$\kappa(L_t)=\pm e^{-\pi i t}\kappa(L)$, so $\kappa^2(L_t)=e^{-2\pi i
  t}\kappa^2(L)$ and $\kappa^2_{\ast}(\{L_t\})=-1$. Thus we see that the 
homomorphism $\kappa^2_{\ast}$ is related to the Maslov index $\mu$ (as
defined, e.g., in~\cite{ALP}) by
$$
   \kappa^2_\ast = -\mu : \pi_1(Gr_{\rm Lag}(V)) \rightarrow \mathbb{Z}.
$$  
Now let $(M,\omega)$ be a symplectic manifold of dimension $2n$. Pick 
a compatible almost complex structure $J$ on $M$  and let $K(M)$ 
be the canonical bundle of $M$, i.e., $K(M):=\Lambda^{(n,0)}T^{\ast}M$ 
is the bundle of $(n,0)$-forms on $M$. Note that
$c_1(K(M))=-c_1(M)$. Let $K^2(M):=K(M)\otimes K(M)$ be the square of
the canonical bundle. \\  
Let $L$ be an immersed Lagrangian submanifold of $M$. For any point $l
\in L$ there is an element of unit length $\kappa(l)$ of $K(M)$ over
$l$, unique up to sign, which restricts to a real volume form on the
tangent space $T_lL$. The squares of these elements give rise to a
section of unit length
$$
\kappa^2_L:L \rightarrow K^2(M).
$$
Now let $F:\Sigma\to M$ be a smooth map with boundary $\partial F$ on
$L$. The {\em symplectic area} of $F$ is 
\[
   \omega(F)=\int_{\Sigma}F^*\omega.
\] 
This defines a map from the relative second homology group to
$\mathbb{R}$, 
$$
   [\omega]: H_2(M,L;\mathbb{Z}) \rightarrow \mathbb{R}.
$$
To define the Maslov class $\mu(F)$, choose a unitary frame for
the tangent bundle $TM$ along $F$. Consider the dual frame and
wedge all its elements. Thus we get a unit 
length section $\kappa_F$ of $K(M)$ over $F$. Now on the boundary
$\partial F=F(\partial\Sigma)$ we also have the section $\kappa^2_L$
defined above. We can uniquely write
$$
   \kappa_L^2 = e^{i\theta} \kappa_F^2
$$
for a function $e^{i\theta}:\partial\Sigma\to S^1$ to the unit
circle. The Maslov class $\mu(F)$ is minus its winding number,
$$
   \mu(F) := \frac{-1}{2\pi} \int_{\partial F}d\theta.
$$
This defines a map
$$
   \mu: H_2(M,L;\mathbb{Z}) \rightarrow \mathbb{Z}.
$$
In view of the discussion above, this definition agrees with the usual
definition of the Maslov class, cf.~\cite{ALP}. 

Now suppose that $c_1(M)=\lambda[\omega]\in H_2(M;\mathbb{R})$. 
It is well-known that if $\partial F$ is trivial in
$H_1(L;\mathbb{R})$, then $F$ represents an element in $[F]\in
H_2(M;\mathbb{R})$ and  
$$
   \mu(F) = 2c_1(M)([F]) = 2\lambda\omega(F).
$$
So in this case $\mu(F)-2\lambda\omega(F)$ depends only on the
boundary $\partial F\in H_1(L;\mathbb{R})$. If, moreover, the map
$H_1(L;\mathbb{R})\to H_1(M;\mathbb{R})$ is trivial, this expression defines a
cohomology class 
$\delta_L\in H^1(L;\mathbb{R})$ via 
$$
   \delta_L(\gamma) := \mu(F) - 2\lambda\omega(F)
$$ 
for some 2-cycle $F$ with $\partial F=\gamma$. 


%%%
\section{proof}
Now assume that $(M,\omega)$ is K\"ahler-Einstein, i.e., $M$ carries a
K\"ahler metric whose Ricci curvature is a multiple of the metric by a
constant $\lambda\in\mathbb{R}$. This is equivalent to saying that the
curvature form of the canonical bundle $K(M)$ equals
$-\frac{2\pi}{i}\lambda\omega$. We denote the connections on $K(M)$
and $K^2(M)$ (induced by the Levi-Civita connection) by $\nabla$. \\ 
Let $L$ be an immersed Lagrangian submanifold of $M$ and let
$\kappa^2_L$ be the canonical section of $K^2(M)$ over $L$ as
above. The section $\kappa^2_L$ defines a connection 1-form $\xi_L$
for $K^2(M)$ over $L$ by the condition $\nabla\kappa^2_L = \xi_L
\otimes \kappa^2_L$.  Since $\kappa^2_L$ has constant length $1$,
$\xi_L$ is an imaginary valued 1-form on $L$. From the Einstein
condition and the fact that $L$ is Lagrangian we get $d(i\xi_L) =
-4\pi\lambda\omega|_L=0$, so the form $i\xi_L$ is closed. \\ 
Let $H$ be the trace of the second fundamental form of $L$ (the mean
curvature vector field of $L$). Thus $H$ is a section of the 
normal bundle to $L$ in $M$ and we have a corresponding 1-form
$\sigma_L:= i_H\omega$ on $L$. The following fact goes back to~
\cite{Oh2} (see also~\cite{Gold1} for a proof): 
$$
   \sigma_L= i \xi_L/2.
$$
(Here the factor $1/2$ is due to the fact that $\xi_L$ is a connection
1-form for $K^2(M)$ rather than $K(M)$.) Thus $\sigma_L$ is a closed
1-form on $L$, called the {\em mean curvature form} on $L$.   

Having explained all the terms in formula (\ref{eq:main}), we now
turn to its proof. Let $F:\Sigma\to M$ be a smooth map from a compact
surface with boundary on $L$. Define the section $\kappa_F$ of $K(M)$
over $F$ as above, using a unitary trivialization of $TM$ over $F$. 
Let $\xi_F$ be the connection 1-form along $F$ defined by 
$\nabla\kappa^2_F = \xi_F \otimes \kappa^2_F$. The Einstein
condition tells us that $d(i\xi_F) = -4\pi\lambda F^*\omega$. Thus by
Stokes' theorem, 
\[
   2\lambda\omega(F)=\int_{\partial F}\frac{-i\xi_F}{2\pi}.
\] 
Recall that along $\partial F$ we have
$\kappa^2_L=e^{i\theta}\kappa^2_F$ for a function
$e^{i\theta}:\partial\Sigma\to S^1$, and the Maslov class is given by 
$$
   \mu(F) = -\frac{-1}{2\pi} \int_{\partial F}d\theta.
$$
The connection 1-forms $\xi_F$ and $\xi_L$ are related by
\[
   \xi_L=\xi_F+i\,d\theta. 
\] 
Thus
\[
   \frac{\sigma_L(\partial F)}{\pi} = \int_{\partial
   F}\frac{i\xi_L}{2\pi} = \int_{\partial F}\frac{i\xi_F}{2\pi} -
   \int_{\partial F}\frac{d\theta}{2\pi} = \mu(F) - 2\lambda\omega(F).  
\] 


%%%
\begin{thebibliography}{99}

\bibitem[Ar]{Ar} A.~Arsie, {\em Maslov class and minimality in
   Calabi-Yau manifolds},  J.~Geom.~Phys.~{\bf 35}, no.~2-3,
   145-156 (2000).

\bibitem[ALP]{ALP} M.~Audin, F.~Lalonde and L.~Polterovich, {\em
     Symplectic rigidity: Lagrangian submanifolds}, In: Holomorphic
   Curves in Symplectic Geometry, ed.~M. Audin and J.~
   Lafontaine, Progress in Math.~{\bf 117}, Birkh\"{a}user, Basel
     (1994). 

\bibitem[D]{D} P.~Dazord, {\em Sur la g\'eometrie des sous-fibr\'es et 
  des feuilletages lagrangiens}, Ann.~Sci.~Ec.~Norm.~Super.~IV,
  Sr.~{\bf 13}, 465-480 (1981).
   
\bibitem[Gold1]{Gold1} E.~Goldstein, {\em A construction of new
  families of minimal Lagrangian submanifolds via torus actions},
  J.~Diff.~Geom.~{\bf 58}, 233-261 (2001).

\bibitem[M]{M} J.-M.~Morvan, {\em Classe de Maslov d'une immersion
  lagrangienne et minimalit\'e}, C.R.~Acad.~Sci.~{\bf 292}, 633-636
  (1981).  

\bibitem[Oh1]{Oh1} Y.-G.~Oh, {\em Floer cohomology of Lagrangian
   intersections and pseudo-holomorphic disks I}, Comm.~Pure Appl.~ 
   Math.~{\bf 46}, no.~7, 949-993 (1993).

\bibitem[Oh2]{Oh2} Y.-G.~Oh,  {\em Mean curvature vector and
  symplectic topology of Lagrangian submanifolds in Einstein-K\"ahler
  manifolds}, Math.~Z.~{\bf 216}, 471-482 (1994). 

%\bibitem[TY]{TY} Thomas, R. P.,  Yau, S.-T. Special Lagrangians,
%  stable bundles and mean curvature flow. Comm. Anal. Geom. 10 (2002),
%  no. 5, 1075--1113. 

\end{thebibliography}
\end{document}




%%% CUT OUT %%%
\section{Oscilation, mean curvature and persistance of Special
  Lagrangians} 
In this section we consider a Calabi-yau manifold
$(M,\omega,\Phi)$. Here $\omega$ is a Calabi-Yau metric and $\Phi$ is
a parallel holomorphic $(n,0)$-form. For an oriented Lagrangian plane
$P$ in the tangent bundle of $M$ we consider the element $\kappa_P$ in
the canonical bundle $K(M)$ which restricts to a volume form on
$P$. We have that $\kappa_P=e^{i \theta} \Phi$ and thus $e^{i \theta}$
is a function on the Grassmanian of oriented Lagrangian planes in
$M$.\\ 
Let $L$ be an oriened Lagrangian submanifold in $M$. We have a phase
function $e^{i \theta}$ on $L$. We'll be interested in the case then
$L$ is exact i.e. $\theta$ is a function $\theta: L \rightarrow
\mathbb{R}$. We define the oscilation of $L$ to be
\[osc(L)=max(\theta)-min(\theta)\] 
Let now $(M,\omega_t,\Phi_t)$ $(0 \leq t \leq 1)$ be a 1-parameter
family of Calabi-Yau structures on $M$ such that all $\omega_t$
represent the same cohomology class in $M$. There are canonical
symplectomorphisms \[\chi_{t_1}^{t_1}:  (M,\omega_{t_1},\Phi_{t_1})
\rightarrow (M,\omega_{t_1},\Phi_{t_2}))\] 
between various elements in the family. We obviously have the following 
\begin{prop}
\label{persist}
For any $\epsilon >0$ there is $\delta >0$ such that the following
holds: Given two times $t_1$ and $t_2$ for which $|t_1-t_2| < \delta$
and an exact Lagrangian submanifold $L$ in $(M, \omega_{t_1})$ the
Lagrangian sumanifold $\chi_{t_1}^{t_1}(L)$ in $(M,\omega_{t_2})$ is
exact and \[|osc(L)-\chi_{t_1}^{t_1}(L)| < \epsilon\] 
\end{prop}
There are various conjectures regarding mean curvature of Lagrangians
with small oscilation (see \cite{TY}). We state the following version 
\begin{conj}
\label{con1}
Given a Calabi-Yau $n$-fold and an $n$-dimensional homology clall
$\eta$ in $M$ there is a constant $c>0$ s.t. for any oriented exact
Lagrangian $L$ in $M$ with $osc(L)<c$ the mean curvature flow of $L$
converges smoothly. 
\end{conj}
Some people conjecture that $c=\pi/2$. We now show that this
conjecture implies a persistance result for Lagrangians in families of
Calabi-Yaus: 
\begin{prop}
Suppose that conjecture \ref{con1} is true. Let $(M,\omega_t,\Phi_t)$
$(0 \leq t \leq 1)$ be a 1-parameter family of Calabi-Yau structures
on $M$ such that all $\omega_t$ represent the same cohomology class in
$M$. Suppose we have a minimal Lagrangian embedding $i_0 :L
\rightarrow (M,\omega_0)$. Then for all times $t$ we have a minimal
Lagrangian embedding $i_t: L \rightarrow (M,\omega_t)$ which is
homotopic to $i_0$.  
\end{prop}
{\bf Proof:} Proposition \ref{persist} and conjecture \ref{con1} imply
that the set of times $t$ for which such $i_t$ exists is both open and
closed. Q.E.D.\\ 
The conclusions of the previous proposition appear to be true for $K3$
surfaces. However we doubt they are true in higher dimensions.    

%%%
The following fact is well known and we give its proof for readers
convenience: 
\begin{lem}
Let $(M,\omega)$ be a symplectic manifold, $L \subset M$ be an
oriented Lagrangian submanifold and $D$ is a disk whose boundary is
mapped to one point $p$ in $L$, i.e. we have a sphere $i:S^2
\rightarrow M$. Then $\frac{I_{\mu}(D)}{2}=c_1(M)(i(S^2))=\lambda
I_{\omega}(D)$.  
\end{lem}
{\bf Proof:} Choose a compatible almost complex structure on $M$ and
let $\kappa_D$ be a section of $K(M)$ over $D$ as before. Let $\kappa$
be a unit length element of $K(M)$ over $p$. On the boundary
$\partial(D)$ we have that $\kappa_D=e^{i \theta} \kappa$ and so
$I_{\mu}(D)$ is twice the winding number of the loop $e^{i
  \theta}$. \\ 
Now suppose that the map $i:S^2 \rightarrow M$ sends the north pole $x
\in S^2$ to $p \in L$. Choose a non-negative cutoff function $\xi$ on
$S^2$ which equals to the distance function to $x$ near $x$ and
$\xi=1$ outside of a neighbourhood of $x$ in $S^2$. The section
$\varphi=\xi \cdot \kappa_D$ is a continuous section of the pullback
of $K(M)$ over $S^2$ with one zero at $x$. So $c_1(M)(i(S^2))$ is the
winding number of $\varphi$ near $p$, which is the winding number of
the loop $e^{i \theta}$. Q.E.D.\\ 
Thus $\frac{I_{\mu}}{2}-\lambda I_{\omega}$ depends just on a homotopy
class of $\partial(D)$ in $L$. Let $K$ be the kernel of the inclusion
homomorphism from $\pi_1(L)$ to $\pi_1(M)$. We see that
$\frac{I_{\mu}}{2}-\lambda I_{\omega}$ is in fact a homomorphism from
$K$ to $\mathbb{R}$. Since $H_1(L,\mathbb{Z})$ is the abelianization
of $\pi_1(L)$ one easily shows that there is a real cohomology class
$\delta_L$ which realizes this homomorphism. 


%%%
A simple explicit example for which the above corollary applies is as
follows: Let $M=\mathbb{C}P^n$. In this case $\lambda=n+1$. For any
positive real number $a$ consider the Lagrangian torus \[L_a=\big(
(z_1,\ldots z_{n+1}) \big| a|z_1|=|z_2|=\ldots =|z_{n+1}| \big) \] 
$L_a$ is minimal only for $a=1$. For each $a$ consider the circle $S_a
\subset L_a$, \[S_a=\big((z_1,1,\ldots,1) \big| |z_1|=\frac{1}{a}
\big)\] 
This circle bounds a holomorphic disk \[D_a=\big((z_1,1,\ldots,1)\big|
|z_1| \leq \frac{1}{a} \big)\] 
The Maslov class of this disk is $2$ and for $a=1$ its area must equal
to $\frac{1}{n+1}$. For $a \neq 1$ its area is different from $
\frac{1}{n+1}$ and so for such $L_a$, $I_{\mu} \neq (n+1)
I_{\omega}$. 


math0310046.extracted_bib
\begin{thebibliography}{99}

\bibitem[Ar]{Ar} A.~Arsie, {\em Maslov class and minimality in
   Calabi-Yau manifolds},  J.~Geom.~Phys.~{\bf 35}, no.~2-3,
   145-156 (2000).

\bibitem[ALP]{ALP} M.~Audin, F.~Lalonde and L.~Polterovich, {\em
     Symplectic rigidity: Lagrangian submanifolds}, In: Holomorphic
   Curves in Symplectic Geometry, ed.~M. Audin and J.~
   Lafontaine, Progress in Math.~{\bf 117}, Birkh\"{a}user, Basel
     (1994). 

\bibitem[D]{D} P.~Dazord, {\em Sur la g\'eometrie des sous-fibr\'es et 
  des feuilletages lagrangiens}, Ann.~Sci.~Ec.~Norm.~Super.~IV,
  Sr.~{\bf 13}, 465-480 (1981).
   
\bibitem[Gold1]{Gold1} E.~Goldstein, {\em A construction of new
  families of minimal Lagrangian submanifolds via torus actions},
  J.~Diff.~Geom.~{\bf 58}, 233-261 (2001).

\bibitem[M]{M} J.-M.~Morvan, {\em Classe de Maslov d'une immersion
  lagrangienne et minimalit\'e}, C.R.~Acad.~Sci.~{\bf 292}, 633-636
  (1981).  

\bibitem[Oh1]{Oh1} Y.-G.~Oh, {\em Floer cohomology of Lagrangian
   intersections and pseudo-holomorphic disks I}, Comm.~Pure Appl.~ 
   Math.~{\bf 46}, no.~7, 949-993 (1993).

\bibitem[Oh2]{Oh2} Y.-G.~Oh,  {\em Mean curvature vector and
  symplectic topology of Lagrangian submanifolds in Einstein-K\"ahler
  manifolds}, Math.~Z.~{\bf 216}, 471-482 (1994). 

%\bibitem[TY]{TY} Thomas, R. P.,  Yau, S.-T. Special Lagrangians,
%  stable bundles and mean curvature flow. Comm. Anal. Geom. 10 (2002),
%  no. 5, 1075--1113. 

\end{thebibliography}