OVERTWISTED ENERGY-MINIMIZING CURL EIGENFIELDS
Robert Ghrist
We consider energy-minimizing divergence-free eigenfields of the curl operator in dimension three from the perspective of contact topology. We give a negative answer to a question of Etnyre and the first author by constructing curl eigenfields which minimize $L^2$ energy on their co-adjoint orbit, yet are orthogonal to an \df{overtwisted} contact structure. We conjecture that $K$-contact structures on $S^1$-bundles always define \df{tight} minimizers, and prove a partial result in this direction.
math0411319.tex
%
% OVERTWISTED PAPER
%
% LAST UPDATED BY RG ON NOV 14 2004
%
\documentclass[11pt]{amsart}
\renewcommand{\baselinestretch}{1.05}

%\usepackage{amsmath,amsthm,epsf,amscd,amssymb,verbatim}
\usepackage{palatino}

%\documentclass[11pt]{article}
\usepackage{latexsym}
\usepackage{amsmath,amsfonts,amssymb,epsfig}
\usepackage{amsthm}
\usepackage{hyperref}


\parskip 0.1in
\parindent 0.0in

%  Set the margins
%\setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in}
%\setlength{\topmargin}{-0.25in}
%\setlength{\textheight}{8.5in}
%\setlength{\textwidth}{6.5in}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{rem}[theorem]{Remark}
\newenvironment{remark}{\begin{rem} \rm}{\end{rem}}
\newtheorem{ide}[theorem]{Idea}
\newenvironment{idea}{\begin{ide} \rm}{\end{ide}}
\newtheorem{exa}[theorem]{Example}
\newenvironment{example}{\begin{exa} \rm}{\end{exa}}
\newtheorem{ques}[theorem]{Question}
\newenvironment{question}{\begin{ques} \rm}{\end{ques}}
\newtheorem{pro}[theorem]{Problem}
\newenvironment{problem}{\begin{pro} \rm}{\end{pro}}
\newtheorem{spe}[theorem]{Speculation}
\newenvironment{specu}{\begin{spe} \rm}{\end{spe}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newcommand{\qed}{{\par\hfill{\tt QED}\smallskip}}
\newcommand{\grad}{\nabla}
\newcommand{\gradtext}{{\mbox{grad}}}
\renewcommand{\div}{\nabla\cdot}
\newcommand{\divtext}{{\mbox{div}}}
\newcommand{\curl}{\nabla\times}
\newcommand{\curltext}{\mbox{curl}}
\newcommand{\vol}{{\mbox{{\bf vol}}}}
\newcommand{\Lie}{{\mathcal L}}
\newcommand{\inv}{^{-1}}
\newcommand{\del}{\partial}
\newcommand{\abs}[1]{\vert #1\vert}
\newcommand{\norm}[1]{\|#1\|}
\font\bbb=msbm10 scaled 1100
\newcommand{\real}{\mbox{\bbb R}}       % {\mbox{\rm I} \hspace{-.03in} {\bf R}}
\newcommand{\comp}{\mbox{\bbb C}}       %{\mbox{\rm I} \hspace{-.09in} {\bf C}}
\newcommand{\hyper}{\mbox{\bbb H}}
\newcommand{\inte}{\mbox{\bbb Z}}
\newcommand{\df}[1]{{\sc #1}}
\newcommand{\bfx}{{\mbox{\bf}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% begin %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\title{OVERTWISTED ENERGY-MINIMIZING CURL EIGENFIELDS}


\author{Robert Ghrist}
\address{Department of Mathematics, University of Illinois,
        Urbana IL, 61801}
\thanks{RG supported in part by NSF PECASE Grant \# DMS - 0337713.}

\author{Rafal Komendarczyk}
\address{Department of Mathematics, Georgia Institute of Technology,
        Atlanta GA, 30332-0160}
\thanks{RK supported in part by NSF Grant \# DMS - 0134408 .}%%%

\subjclass{Primary: 58J05, 37J55; Secondary: 53D10}
\keywords{eigenfield; contact structure; overtwisted}

\begin{abstract}
We consider energy-minimizing divergence-free eigenfields of the
curl operator in dimension three from the perspective of contact
topology. We give a negative answer to a question of Etnyre and the
first author by constructing curl eigenfields which minimize $L^2$
energy on their co-adjoint orbit, yet are orthogonal to an
\df{overtwisted} contact structure. We conjecture that $K$-contact
structures on $S^1$-bundles always define \df{tight} minimizers, and
prove a partial result in this direction.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

Eigenfields of the curl operator form an important class of
solutions to the steady Euler equations in dimension three. These
equations model the velocity field of an inviscid, incompressible
fluid flow on a Riemannian manifold \cite{Khesin98}. It has been
observed \cite{Hamilton85,Chicone,Ghrist00_2} that there is a
correspondence between curl eigenfields and contact 1-forms in
dimension three. Recall that a contact 1-form
$\alpha\in\Omega^1(M^3)$ is one which satisfies $\alpha\wedge
d\alpha\neq 0$. Such 1-forms have as their kernel a totally
nonintegrable plane field known as a {\em contact structure}. The
correspondence is this: any nonvanishing curl eigenfield is dual
to a contact 1-form via the Riemannian metric; and any contact
1-form can be realized as dual to a nonvanishing curl eigenfield
for some Riemannian structure.

This observation raises interesting questions concerning the
interplay between fluid dynamical properties of curl eigenfields and
topological properties of contact structures. More specifically, one
can investigate how the topological {tight/overtwisted} dichotomy
for contact structures relates to the physical properties of the
fluid like {energy}, {periodic orbits} etc. In \cite{Ghrist00_2} (p.
16) the authors asked if variational principles of the fluid flows
descend to variational principles for the corresponding contact
structures.

\begin{question}\label{qe:q1}
Does a nonvanishing curl eigenfield which minimizes ($L^2$) energy
on its coadjoint orbit under the volume-preserving diffeomorphism
group necessarily define a tight contact structure?
\end{question}

Using a recent result of the second author
\cite{Komendarczyk_nodal}, we give a negative answer to this
question. This involves constructing special $S^1$-invariant curl
eigenfields on products $S^1\times\Sigma$, where $\Sigma$ is a
closed orientable surface of genus $g>0$. In the second part of this
note, we demonstrate that under additional symmetry conditions, the
energy-minimization condition does yield a tightness constraint on
the associated contact structures.

Throughout the paper, we use the language of differential forms
and global analysis, as in \cite{Khesin98}. We restrict attention
to the class of volume-preserving vector fields and 1-forms
exclusively.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Contact structures}

Let $(M,g)$ be a Riemannian 3-manifold. The \df{curl operator} on
1-forms is $\ast\,d:\Omega^1(M)\rightarrow\Omega^1(M)$, where $\ast$
is the Hodge star operator and $d$ the exterior derivative. An
\df{eigenform} of curl is any 1-form $\alpha$ satisfying
$\ast\,d\alpha=\mu\alpha$ for some $\mu\in\real$.

Given a nonvanishing curl eigenform $\alpha$ with nonzero eigenvalue
$\mu\neq 0$, it easily follows that
\[
    \alpha \wedge d\alpha = \alpha \wedge (\mu\ast\alpha) \neq 0.
\]
Therefore, $\alpha$ is a \df{contact form} and the plane field
$\xi=\text{ker}(\alpha)$ defines a {\it contact structure}
--- a nowhere integrable plane field on $M$. If $\alpha$ is the dual
1-form to a vector field, then $\xi$ is the orthogonal plane field.
All contact structures in this paper are transversely orientable, as
they are dual to globally-defined vector fields.

It has been known since the work of Bennequin and Eliashberg
\cite{Benn,Eli} that there are two fundamentally different classes
of contact structures.

\begin{definition}\label{def:tight_overt}
A contact structure $\xi$ is \df{overtwisted} if and only if there
exists an embedded disk $D^2\subset M$ such that $D$ is transverse
to $\xi$ near $\partial D$ but $\partial D$ is tangent to $\xi$. Any
contact structure which is not overtwisted is called \df{tight}.
\end{definition}

As one might expect from the definition, it is rather difficult to
determine if a given contact structure is tight or overtwisted. One
of the more successful recent techniques for solving the
classification problem involves examining the \df{characteristic
surface}.
\newpage
\begin{definition}
Let $X$ be a vector field preserving the contact plane distribution
$\xi$, i.e., $\Lie_X\xi=0$.
The characteristic surface $\Gamma_X$ is the set of tangencies of $X$
with $\xi$,
\begin{gather}\label{eq:char_surface}
    \Gamma_X=\{p\in M: X_p\in \xi_p\}.
\end{gather}
\end{definition}

The following result is essential for our study of $S^1$-invariant
curl eigenforms on circle bundles.

\begin{theorem}[Giroux \cite{Giroux01}]\label{th:giroux_bundle}
Let $\xi$ be an $S^1$-invariant contact structure on a principal
circle bundle $\pi:P\to\Sigma$ over a closed oriented surface
$\Sigma$. Let $\Gamma=\pi(\Gamma_{S^1})$ be a projection of the
characteristic surface $\Gamma_{S^1}$ onto $\Sigma$. Denote by
$e(P)$ be the Euler number of the bundle $P$. Then $\xi$ and all
covers of $\xi$ are tight if and only if one of the following holds:
\begin{itemize}
    \item[(i)] For $\Sigma\neq S^2$ none of the connected
        components of $\Sigma/\Gamma$ is a disc.
    \item[(ii)] For $\Sigma=S^2$, $e(P)<0$ and $\Gamma=\emptyset$.
    \item[(iii)] For $\Sigma=S^2$, $e(P)\ge 0$ and $\Gamma$ is connected.
   \end{itemize}
\end{theorem}
A contact structure all of whose covers are tight is called
\df{universally tight}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Energy and eigenvalues}

An important feature of any curl eigenform $\alpha$ is the fact that
it extremizes the $L^2$-energy, defined as,
\begin{gather}\label{eq:energy}
E(\alpha)=\|\alpha\|^2_{L^2}=\int_M \alpha\wedge\ast\alpha,
\end{gather}
among all 1-forms obtained from $\alpha$ by pullbacks through volume
preserving diffeomorphisms. The set of such forms, $\Xi_\alpha$, is
the \df{coadjoint orbit} of $\alpha$ under the action of the
volume-preserving diffeomorphism group of $M$:
\begin{gather}
    \Xi_\alpha = \{\beta: \beta=\varphi_\ast(\alpha),\ \varphi\in
    \text{Diff}_0(M),\ \varphi_\ast(\ast 1)=\ast 1\}.
\end{gather}

The question of energy minimization on the coadjoint orbit is more
delicate, and closely related to spectral data. The following result
is one of the few general results available:

\cite{Khesin98}:
\begin{proposition}\label{th:prop_minimizers}
A curl eigenform $\alpha_1$, (i.e. an eigenform of the curl operator
$\ast\,d:\Omega^1(M)\to \Omega^1(M)$) corresponding to the first
eigenvalue $\mu_1\neq 0$ is a minimizer of the energy $E$ on
$\Xi_{\alpha_1}$.
\end{proposition}
\begin{proof}
The operator $\ast\,d$ is unbounded, closed, self-adjoint, and
elliptic; it has a compact inverse $\ast\,d^{-1}$ defined on the
orthogonal complement of its kernel (see \cite{Yoshida90}).  We can
also choose an orthonormal basis of eigenforms $\{\alpha_i\}$ in
$(L^2(M,\Lambda^1 T^\ast M),(\cdot,\cdot)_{L^2})$ such that,
\begin{gather}\label{eq:eigenbeltrami}
\ast\,d^{-1}\alpha_i=\frac{1}{\mu_i}\,\alpha_i,\qquad 0<\mu^2_1\leq
\mu^2_2\leq\ldots\leq \mu^2_i\leq\ldots
\end{gather}
For an arbitrary $L^2$ 1-form $\beta\in \text{Im}(\delta)$ we have
\begin{gather}\label{eq:helicity_estim}
%\begin{array}{rcl}
 |(\ast\,d^{-1}\beta,\beta)_{L^2}|=|\sum_i
\frac{1}{\mu_i}(\alpha_i,\beta)^2_{L^2}|
\leq\frac{1}{|\mu_1|}(\beta,\beta)_{L^2}=\frac{1}{|\mu_1|}E(\beta).
%\end{array}
\end{gather}
One obtains a lower bound for the energy $E(\beta)$,
\begin{gather*}
E(\beta)\ge |\mu_1||(\ast\,d^{-1}\beta,\beta)_{L^2}|.
\end{gather*}
The above inequality becomes the equality if and only if
$\beta=\alpha_1$. The claim follows from the fact that the
\df{helicity}, $(\ast\,d^{-1}\beta,\beta)_{L^2}$, is invariant under
volume preserving transformations see \cite{Khesin98}.
\end{proof}

Since the curl operator is a formally self-adjoint first-order
elliptic operator of the first order, the principal eigenvalue
$\mu_1$ enjoys the variational characterization through the
Rellich's quotient. As a corollary of the following considerations
and Lemma \ref{th:lemm_chander} we have,
\begin{gather*}
    \mu_1 = \inf_{\alpha\in \mathcal{H}}
    \frac{|(\ast\,d\alpha,\alpha)_{L^2}|}{\|\alpha\|^2_{L^2}}
    \quad
    \Rightarrow\quad \mu^2_1=\inf_{\alpha\in \mathcal{H}}
    \frac{(\Delta^1_M\alpha,\alpha)_{L^2}}{\|\alpha\|^2_{L^2}},
\\
    \mathcal{H}=\text{Ker}(\ast\,d)^\perp=
    \{\alpha\in\Omega^1(M):\alpha=\delta\beta,
    \text{ for some }\beta\in\Omega^2(M)\},\
\end{gather*}
Indeed, it follows that the curl squared is equal to the Hodge
Laplacian , $(\ast\,d)^2=\delta\,d$, on $\mathcal{H}$. Therefore any
curl eigenform $\alpha$, (i.e. $\ast\,d\alpha=\pm\mu\,\alpha$) is
automatically a co-closed $\mu^2$-eigenform of the Hodge Laplacian
i.e.
\begin{gather}\label{eq:betrami-lap}
\Delta^1_M\alpha=\delta\,d\,\alpha=\ast\,d\ast d\alpha=\mu^2\,\alpha,
\end{gather}
Clearly the curl $\ast\,d$ commutes with the Hodge Laplacian
$\Delta^1_M$, therefore both of these operators are simultaneously
diagonizable on $\mathcal{H}$ in a suitable orhonormal basis of curl
eigenforms;
\begin{gather*}
\mathcal{H}=\bigoplus^\infty_{i=1} E^{\Delta} (\mu^2_i), \quad
E^{\Delta}(\mu^2_i)\perp E^{\Delta}(\mu^2_j),\ i\neq j
\qquad0<\mu^2_1\leq \mu^2_2\leq\ldots\leq \mu^2_i\leq\ldots
\end{gather*}
where $E^{\Delta}(\mu^2_i)$ stands for the $\mu^2_i$-eigenspace of
$\Delta^1_M$, and
\begin{gather*}
E^{\Delta}(\mu^2_i)=E^{\ast\,d}(\mu_i)\oplus E^{\ast\,d}(-\mu_i).
\end{gather*}
(We allow one of $E^{\ast\,d}(\mu_i)$, $E^{\ast\,d}(-\mu_i)$ to be
trivial.) We may conclude further that there exist two positive
operators $\sqrt{\Delta}_{+}$, $\sqrt{\Delta}_{-}$, such that
$\ast\,d=\sqrt{\Delta}_{+}-\sqrt{\Delta}_{-}$.

The following useful fact, which can be traced back to work in
\cite{Chandrasekhar57}, tells us how to effectively find a basis of
curl eigenforms from a basis of co-closed $\Delta^1_M$-eigenforms.
\begin{lemma}\label{th:lemm_chander}
Any curl $\mu$-eigenform is automatically a co-closed
$\mu^2$-eigenform of the Laplacian $\Delta^1_M$. Conversely, given a
co-closed $\mu^2$-eigenform $\alpha\in \Omega^1(M)$ of $\Delta^1_M$
there exists a corresponding $\pm\mu$-curl eigenform $\beta\in
\Omega^1(M)$ given by
\begin{gather}\label{eq:ck_beltrami}
\beta=\mu\,\alpha\pm\ast\,d\alpha
\end{gather}
\end{lemma}
\begin{proof}
We verify this claim by a direct calculation. Since
$\delta\alpha=0$, and $\ast d\ast d
=\delta\,d=\Delta|_{\mathcal{H}}$, $\Delta^1_M\alpha=\mu^2\alpha$,
$\delta\alpha=0$ we obtain,
\begin{gather*}
\ast\,d\beta=\mu\ast d\alpha\pm\Delta^1_M\alpha=\pm\mu^2\ast
\alpha+\mu\ast d\alpha=\pm\mu\,\beta
\end{gather*}
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Overtwisted principal eigenfields}

For the purpose of producing an overtwisted principal curl eigenform
we assume that $M$ is a trivial bundle $P=S^1\times\Sigma$ and the
metric $g$ on $P$ is a product metric $g=1\oplus g_\Sigma$ such that
fibers are of constant length $l$. We will construct our example
using a sequence of lemmas.

In the case of the product $(P, g)$, $P=S^1\times\Sigma$, $g=1\oplus
g_\Sigma$, the space of smooth 1-forms $\Omega^1(P)$ decomposes with
respect to the $L^2$-inner product induced by the metric $g$ as,
\begin{gather}\label{eq:tan_norm}
\Omega^1(P)=\Omega^1_N(P)\oplus\Omega^1_T(P)
\end{gather}
where,
\begin{gather*}
\Omega^1_N(P)=\{\alpha\in \Omega^1(P): \alpha=f\eta,\ f\in C^\infty(P)\},
\\
\Omega^1_T(P)=\Omega^1_N(P)^\perp\cap\Omega^1(P),
\\
\Omega^1_T(P)=\{\alpha\in \Omega^1(P): \alpha(X_\eta)=0\}
\end{gather*}
The following lemma is another consequence of the product metric
assumption.

\begin{lemma}\label{th:dec_lap}
The Laplacian $\Delta^1_P$ preserves $\Omega^1_N(P)$,
$\Omega^1_T(P)$ and for $\alpha=f\eta+\beta$, $f\eta\in
\Omega^1_N(P)$, $\beta \in \Omega^1_T(P)$ we have the following
formula for the Laplacian,
\begin{gather}\label{eq:lap-product}
    \Delta^1_P\alpha=(\Delta^0_P f)\eta +
    \Delta^1_P\beta=(-\Lie^2_\eta+\Delta^0_\Sigma f)\eta +
    (-\Lie^2_\eta\,\beta+\Delta^1_\Sigma\,\beta)
\end{gather}
\end{lemma}

\begin{proof}
The first claim immediately follows from the formula
\eqref{eq:lap-product}, and the formula itself from the general form
of the product Laplacian on $k$-forms:
$\Delta_{S^1\times\Sigma}^k=\sum_j
\Delta_{S^1}^{j}\otimes\text{Id}+\text{Id}\otimes\Delta^{k-j}_{\Sigma}$
(see, e.g., \cite{Rosenberg97}).
\end{proof}
%\begin{corollary}
%For a curl eigenform $\alpha$ on the product manifold $(P,g)$ (not
%necessarily $X_\eta$-invariant) with eigenvalue $\mu$, the
%$\eta$-component $f=\alpha(X_\eta)$ is an $\mu^2$-eigenfunction of
%the Laplacian.
%    \begin{gather}
%        \Delta^0_P f=\mu^2\,f
%    \end{gather}
%\end{corollary}
%
%\begin{proof}
%The proof is an immediate consequence of \eqref{eq:betrami-lap} and
%\eqref{eq:lap-product}.
%\end{proof}
%
\begin{lemma}\label{th:eigenvalues_lemma}
On the product manifold $P=S^1\times \Sigma$, $g=1\oplus g_\Sigma$
with constant length $l$ fibres. The first eigenvalue $\mu_1$ of the
curl operator satisfies,
\begin{gather*}
    \mu^2_1=\min\left\{\nu_1,\Bigl(\frac{2\pi}{l}\Bigr)^2\right\},
    \quad\text{where}\quad
    \nu_1=\inf_{f\in L^2(\Sigma),\
    f\neq\text{const}}\left\{\frac{(\Delta^0_\Sigma
    f,f)_{L^2}}{\|f\|^2_{L^2}}\right\}.
\end{gather*}
\end{lemma}

\begin{proof}
From the decomposition \eqref{eq:tan_norm} and the fact that
$\Delta^1_P$ preserves $\Omega^1_T(P)$ and $\Omega^1_N(P)$ (see
Lemma \ref{th:dec_lap}) we have
\begin{gather}\label{eq:eigenvalues}
    \mu^2_1=\min\left\{\mu^2_{1,T}, \mu^2_{1,N}\right\};\quad
    \mu^2_{1,r}=\inf_{\alpha\in \mathcal{H}\cap
    \Omega^1_r(P)}\left\{\frac{(\Delta^1_P\alpha,\alpha)_{L^2}}{\|\alpha\|^2_{L^2}}\right\}\
    \ r=T,N.
\end{gather}
In order to calculate $\mu^2_{1,N}$ notice that for any $\alpha\in
\mathcal{H}\cap\Omega^1_N(P)$, $\alpha=f\,\eta$, the function $f$ is
constant on the fibres; hence $f\in C^\infty(\Sigma)$. Indeed,
$\delta\alpha=0$, and, since $\nabla\eta=0$ in the adapted frame
$X_\eta=\{X_1,X_2,X_3\}$, we obtain
\begin{gather*}
0=\delta\alpha=\iota(X_i)\nabla_i\alpha=\iota(X_i)(\nabla_i f\eta+f
\nabla_i\eta)=\nabla_1 f=X_\eta f.
\end{gather*}
We conclude now that for any
$\alpha\in\mathcal{H}\cap\Omega^1_N(P)$, $\Delta^1_\Sigma
\alpha=(\Delta^0_\Sigma f)\eta$ and
\begin{gather*}
    (\Delta^1_P\alpha,\alpha)_{L^2}=(\Delta^0_\Sigma f\eta,\eta)_{L^2}
    =\int_P (f\,\Delta^0_\Sigma f)\eta\wedge\ast\eta
    =(\Delta^0_\Sigma f,f)_{L^2},
\end{gather*}
where $\eta\wedge\ast\eta=\ast 1$. Therefore
\begin{gather}\label{eq:eigenvalue_N}
    \mu^2_{1,N}=\nu_1,\qquad \nu_1=\inf_{f\in L^2(\Sigma),\
    f\neq\text{const}}\left\{\frac{(\Delta^0_\Sigma
    f,f)_{L^2}}{\|f\|^2_{L^2}}\right\}.
\end{gather}
In other words $\mu^2_{1,N}$ is equal to the first eigenvalue of the
scalar Laplacian $\Delta^0_\Sigma$ on $\Sigma$.

In order to calculate $\mu^2_{1,T}$ we first calculate the
orthogonal basis of eigenforms of $\mathcal{H}\cap\Omega^1_T(P)$.
Let $\{\beta_m\}$ be an orthonormal basis of $\Delta^1_\Sigma$-
eigenforms on $\text{Ker}(\Delta^1_\Sigma)\oplus
\text{Im}(\delta)\subset L^2(\Lambda^1 T^\ast \Sigma)$, define for
all $m, n\in \mathbb{Z}^+$:
\begin{gather}\label{eq:eigen-basis}
h_0=g_0=1,\quad h_n=\cos\bigl(\frac{2\pi n t}{l}\bigr),\ g_n=\sin\bigl(\frac{2\pi n
t}{l}\bigr),
\\
%g_n,h_n\in C^\infty(S^1),\ n\in\mathbb{Z}^+,\\
\notag \alpha^g_{n\,m}=g_n\,\beta_m,\qquad\text{ and }
\qquad\alpha^h_{n\,m}=h_n\,\beta_m .
%\quad n,m\in\mathbb{Z}^+.
\end{gather}
Clearly $\{\alpha^g_{n\,m},\alpha^h_{n\,m}\}$ is a set of
1-eigenforms of $\Delta^1_P$ on $\mathcal{H}\cap\Omega^1_T(P)$.
Indeed, from \eqref{eq:lap-product} we have,
\begin{gather*}
\Delta^1_P\alpha^r_{mn}=\gamma^r_{mn}\alpha^r_{mn},\quad
\gamma^r_{mn}=\Bigl(\frac{2\pi n}{l}\Bigr)^2+\tilde{\nu}_m
\end{gather*}
where $\tilde{\nu}_m$ is the $m$-th eigenvalue of $\Delta^1_\Sigma$.
One easily shows that $\{\alpha^g_{n\,m},\alpha^h_{n\,m}\}$ is an
orthonormal basis of $\mathcal{H}\cap\Omega^1_T(P)$.
Consequently, all eigenforms of $\Delta^1_P$ on
$\mathcal{H}\cap\Omega^1_T(P)$ are listed in \eqref{eq:eigen-basis},
and we have
\begin{gather}\label{eq:eigenvalue_T}
\mu^2_{1,T}=\min\left\{\Bigl(\frac{2\pi}{l}\Bigr)^2,\tilde{\nu}_1\right\}
\end{gather}
It is left to show that $\tilde{\nu}_1=\nu_1$. By the Hodge
decomposition theorem:
\begin{gather*}
\Omega^1(\Sigma)=
\text{Ker}(\Delta^1_\Sigma)\oplus\text{Im}(d_\Sigma)
\oplus\text{Im}(\delta_\Sigma).
\end{gather*}
Moreover, $\Omega^0(\Sigma)\simeq\Omega^2(\Sigma)$ through the
Hodge-star isometry. We conclude that
$\text{Im}(\delta_\Sigma)=\{\ast\,d_\Sigma f;f\in
C^\infty(\Sigma)\}$. Since $\Delta^1_\Sigma$ commutes with
$\ast\,d_\Sigma$, any $\nu_m$-eigenfunction $f_m$ results in a
$\nu_m$-eigenform $\ast\,d_\Sigma f_m$. Therefore
$\tilde{\nu}_1=\nu_1$ and the lemma follows from
\eqref{eq:eigenvalues}, \eqref{eq:eigenvalue_N}, and
\eqref{eq:eigenvalue_T}.
\end{proof}

In \cite{Komendarczyk_nodal} the following theorem was proved:

\begin{theorem}[\cite{Komendarczyk_nodal}]\label{th:payne}
For an arbitrary closed compact orientable surface $\Sigma$, there
exists a smooth metric $g_\Sigma$ such that a nodal set
$f^{-1}_1(0)$ of the principal eigenfunction $f_1$ of
$\Delta^0_\Sigma$ is a single embedded circle which bounds a disc in
$\Sigma$.
\end{theorem}

Combining the above theorem with Lemma \ref{th:eigenvalues_lemma}
results in the following,

\begin{theorem}\label{th:principal_overt}
Let $\Sigma\neq S^2$ be an orientable surface of an arbitrary
nonzero genus. One can prescribe a metric $g_\Sigma$ on $\Sigma$
such that there exist an overtwisted curl eigenfield $v$ on the
product manifold $(S^1\times\Sigma,1\oplus g_\Sigma)$ which
minimizes the energy \eqref{eq:energy} on the coadjoint orbit
$\Xi_\alpha$.
\end{theorem}

\begin{proof}
In the first step we choose a metric $g_\Sigma$ on $\Sigma\neq S^2$
constructed in Theorem \ref{th:payne} and assume that the length of
fibres in $(S^1\times\Sigma,1\oplus g_\Sigma)$ is equal to $l$. By
Lemma \ref{th:eigenvalues_lemma} we may choose $l$ small so that the
first eigenvalue satisfies $\mu_1=\nu_1$. By earlier considerations
$E^\Delta(\mu^2_1)=E^{\ast\,d}(\mu_1)\oplus E^{\ast\,d}(-\mu_1)$ and
$E^\Delta(\mu^2_1)$ is spanned by two independent $\pm\mu_1$- curl
eigenforms defined by Lemma \ref{th:lemm_chander} as
\begin{gather*}
\alpha_{\pm}=f_1\eta\pm\ast_{\Sigma}\,d\,f_1.
\end{gather*}
The dimension is two since $g_\Sigma$ can be chosen in a residual
subset of an open set of metrics: see, \cite{Komendarczyk_nodal}.
These forms are nonvanishing since the set of zeros is clearly
equal to the singular part of the nodal set of $f_1$. This
singular part is empty for a generic choice of metric (see
\cite{Uhlenbeck76}). Both forms are $S^1$-invariant and
overtwisted by Theorem \ref{th:giroux_bundle}. Indeed, the
projection of the characteristic surface $\Gamma_{S^1}$ of
$\alpha_{\pm}$ onto $\Sigma$ is equal to $f^{-1}_1(0)$, the nodal
set of $f_1$. By the choice of the metric $\pi(\Gamma_{S^1})$
bounds a disk. Now, the dual curl eigenfields $\alpha^{\#}_{\pm}$
minimize energy \eqref{eq:energy} on $\Xi_{\alpha_{\pm}}$ due to
Proposition \ref{th:prop_minimizers}.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Symmetry and tight energy minima}

There are certain cases for which principal curl eigenforms can
never be overtwisted. Courant's theorem on nodal sets quickly yields
such a rigidity result for $S^2\times S^1$:

\begin{proposition}
Let $M=S^2\times S^1$ with any product metric giving the $S^1$
fibers a constant length $l$. Then the principal eigenform of curl
on $M$ is never overtwisted.
\end{proposition}
\begin{proof}
It follows from Lemma \ref{th:eigenvalues_lemma} that if the first
eigenvalue $\mu_1$ is equal to $\frac{2\pi}{l}$ then the curl
eigenforms have zeros (since the dual vector fields are tangent to
$S^2$). If $\mu_1=\nu_1$ Theorem \ref{th:giroux_bundle} tells us
that a contact form is tight if and only if the projection of
$\Gamma_{S^1}=f_1^{-1}(0)$ is a single circle. On the other hand
Courant's theorem implies that the principal eigenfunction on a
closed surface always has exactly two nodal domains, which in turn
implies tightness.
\end{proof}

For the more general case, tightness can be forced by additional
symmetry. Let $P$ be a principal $S^1$-bundle over a closed
orientable surface $\Sigma$, equipped with a bundle metric of
constant length fibres. Assume that all fibers are geodesics, the
bundle metric is invariant under the action of a Killing field $X$
tangent to the fibres, and the projection $\pi:P\rightarrow \Sigma$
is a Riemannian submersion. Cartan's equations of structure imply
that dual 1-form $\eta=X^\flat$ satisfies
\begin{gather}
\ast\,d\eta=2 \lambda\,\eta,\qquad \lambda\in C^\infty(P).
\end{gather}
Therefore, $\eta$ defines a curl eigenform, if
$\lambda(x)=\lambda=\text{const}$. It is shown in
\cite{Nicolaescu98} that $\lambda=-e(P)/\gamma$, where
$\gamma=\|X\|_p$ and $e(P)$ is an Euler number of the bundle. So
called Sasakian structures are special case of such curl eigenforms
and occur in the case $\lambda=1$,  Hopf fields on the round $S^3$
are the classical example (see \cite{Khesin98}). Such structures are
called \df{$K$-contact},\cite{Blair02}, structures
%, or $(K,\lambda)$-contact structures
(see \cite{Nicolaescu98}). By Theorem \ref{th:giroux_bundle} $\eta$
is necessarily tight since the contact plane distribution is
orthogonal to the fibers.

Denote by $\mathcal{H}_{S^1}$ the subspace of $S^1$- invariant
1-forms in $\mathcal{H}\subset\Omega^1(P)$.
\begin{proposition}\label{th:K-cont_min}
Any curl eigenform $\eta$ defined by a $K$-contact structure is
always energy-minimizing on $\mathcal{H}_{S^1}\cap\Xi_\eta$. Let
$\nu$ be the first nonzero eigenvalue of the scalar Laplacian on
$\Sigma$. If $\nu>3\lambda^2$ then $\eta$ is a principal
curl eigenform on $\mathcal{H}_{S^1}$.
\end{proposition}

\begin{proof}
We provide the proof of the first claim for $\lambda>0$ (in the case
of $\lambda<0$ the reasoning is analogous). The space
$\mathcal{H}_{S^1}$ decomposes as
\begin{gather*}
\mathcal{H}_{S^1}=\mathcal{H}^+_{S^1}\oplus\mathcal{H}^-_{S^1},
\end{gather*}
where $\mathcal{H}^\pm_{S^1}$ is a subspace spanned by
positive/negative curl eigenforms. We need to show that $\eta$ is an
energy minimizer on $\mathcal{H}_{S^1}\cap\Xi_\eta$. Given a volume
preserving diffeomorphism $\varphi:P\rightarrow P$ we denote
$\eta_\varphi=\varphi_\ast(\eta)\in \Xi_\eta$. Under the assumptions
on the $\varphi$ action, $\eta_\varphi\in\mathcal{H}_{S^1}\cap\Xi_\eta$.
We expand $\eta_\varphi$ in the eigenbasis of
curl eigenforms \eqref{eq:eigenbeltrami}, $\eta_\varphi=\sum_{i\ge
0} c^{+}_i\alpha^{+}_i+\sum_{i< 0} c^{-}_i\alpha^{-}_i$, where
$\{\alpha^{\pm}_i\}$ span $\mathcal{H}_{S^1}$. Since the
helicity $(\ast d^{-1}\eta_\varphi,\eta_\varphi)$ is invariant under
$\varphi$, as in \eqref{eq:helicity_estim}, we obtain
\begin{gather*}
0<\frac{E(\eta)}{2\lambda}=(\ast d^{-1}\eta,\eta)=(\ast
d^{-1}\eta_\varphi,\eta_\varphi)=\sum_{i\ge 0}
\frac{(c^{+})^2}{\mu^{+}_i}+\sum_{i< 0} \frac{(c^{-})^2}{\mu^{-}_i}
\end{gather*}
where $\mu^\pm_i$, positive/negative eigenvalues of $\ast d$ on
$\mathcal{H}_{S^1}$. Since the second sum is negative we can
estimate $\mu^+_1 (\ast d^{-1}\eta,\eta)\leq E(\eta_\varphi)$. To
finish the proof it suffices to show that $2\lambda=\mu^+_1$, then we
obtain $E(\eta)\leq E(\eta_\varphi)$ which proves the claim.

We derive the equality $2\lambda=\mu^+_1$ by a calculation in the
adapted orthonormal coframe $\{\eta^1=\gamma\,\eta,\eta^2,\eta^3\}$ dual to $\{X_1=\frac{1}{\gamma}X,X_2,X_3\}$, where $X$ is the Killing field tangent to the fibers and $\ker{\alpha_1}=\text{span}\{X_2,X_3\}$. Let
$\alpha_1=a_i\eta^i=f\eta+\beta$ be the curl eigenform satisfying
\begin{gather}\label{eq:alpha_1}
 \ast d\alpha_1=\mu^+_1\alpha_1.
\end{gather}
Since, $\Lie_X\alpha_1=0$, using the Cartan formula we obtain:
 \begin{eqnarray*}
 0 & = & \Lie_{X} \alpha = \mu^+_1\,\iota(X)\ast\alpha + df\\
-df=-\nabla_i f\,\eta^i & = & \mu^+_1\,\gamma\,\iota(X_1)\ast\alpha\\
df=\nabla_1 f\,\eta^1+\nabla_2 f\,\eta^2+\nabla_3 f\,\eta^3 &
= & \mu^+_1\,\gamma\,(a_2\,\eta^3-a_3\,\eta^2).
 \end{eqnarray*}
This leads to the following equations,
 \begin{gather}\label{eq:eq_Lie}
    \nabla_1 f=0,\quad
    \nabla_2 f=-\mu^+_1\,\gamma\,a_3,\quad
    \nabla_3 f=\mu^+_1\,\gamma\,a_2.
\end{gather}
Applying \eqref{eq:alpha_1} and \eqref{eq:eq_Lie} to
$d\,\alpha=\nabla_i a_k\eta^i\wedge\eta^k -
a_k\Gamma^k_{ji}\eta^i\wedge\eta^j$ (where $\Gamma^k_{ji}$ are the
Christoffel symbols in the frame $\{\eta^i\}$) leads to,
\begin{gather}\label{eq:der1}
\frac{\mu^+_1}{\gamma}\, f=\frac{1}{\mu^+_1\,\gamma}(-\nabla_{2 2}
f-\nabla_{33}f+\Gamma^2_{33}\nabla_2 f+\Gamma^3_{22}\nabla_3 f)+
\frac{2\lambda}{\gamma}f.
\end{gather}
Consequently we obtain the following equation for $f$:
\begin{gather}\label{eq:scalar_lap}
 \Delta^0_{P}f=\mu^+_1(\mu^+_1-2\lambda)f.
\end{gather}
Equations \eqref{eq:eq_Lie} imply that for $\alpha_1$ to be
nontrivial, $f$ cannot be a constant zero function. Since
$\Delta^0_{P}$ is a positive operator we conclude that
$\mu^+_1\geq 2\lambda$; consequently, $\mu^+_1=2\lambda$ because
$\mu^+_1$ is the first positive eigenvalue. The equation
\eqref{eq:scalar_lap} is valid for any $S^1$-invariant
$\mu$-eigenform. Thus the proof of the second statement follows
from the equation $\nu=\mu(\mu-2\lambda)$.
\end{proof}

We may think about $\lambda\neq 0$ as a ``topological deviation''
from the $\lambda=0$ case. Therefore $\eta$ is the most natural
candidate for the principal curl eigenform, or the energy
minimizer on $\Xi_\eta$. We note that Hopf fields are principal
curl eigenfields and therefore energy minimizers. Lemma
\ref{th:eigenvalues_lemma} and Proposition \ref{th:K-cont_min}
show that in general we expect the first eigenvalue $\mu_1$ to
depend on the length of the fibre $l$, the spectrum on the surface
and $\lambda$. Therefore $\eta$ may not always be a principal curl
eigenform.
%We may conjecture, however, that it is always a minimizer on $\Xi_\eta$.

\begin{conjecture}
The curl eigenform $\eta$ defined by a $K$-contact structure is
always energy-minimizing on $\Xi_\eta$.
\end{conjecture}

\small

\section{Acknowledgments}
The second author would like to thank Jason Cantarella for the
inspiring discussion during Georgia Topology Conference in Athens
2000.

%\bibliography{phdbib}

\bibliographystyle{plain}

\begin{thebibliography}{10}

\bibitem{Khesin98}
V.~Arnold and B.~Khesin.
\newblock {\em Topological Methods in Hydrodynamics}, volume 125 of {\em
  Applied Mathematical Sciences}.
\newblock Springer-Verlag, New York, 1998.

\bibitem{Benn}
D. Bennequin.
\newblock Entrelacements et \'{e}quations de Pfaff.
\newblock {\em Ast\'{e}risque}, 87--161, Soc. Math. France, Paris, 1983.

\bibitem{Blair02}
D.~Blair.
\newblock {\em Riemannian geometry of contact and symplectic manifolds}, volume
  203 of {\em Progress in Mathematics}.
\newblock Birkh\"auser Boston Inc., Boston, MA, 2002.

\bibitem{Chandrasekhar57}
S.~Chandrasekhar and P.~Kendall.
\newblock On force-free magnetic fields.
\newblock {\em Astrophys. J.}, 126:457--460, 1957.

\bibitem{Chicone}
C. Chicone.
\newblock The topology of stationary curl parallel solutions of
Euler's equations.
\newblock {\em Israel J. Math.} 39:1-2, 161--166, 1981.


\bibitem{Hamilton85}
S.~S. Chern and R.~S. Hamilton.
\newblock On {R}iemannian metrics adapted to three-dimensional contact
  manifolds.
\newblock In {\em Workshop Bonn 1984 (Bonn, 1984)}, volume 1111 of {\em Lecture
  Notes in Math.}, pages 279--308. Springer, Berlin, 1985.

\bibitem{Eli}
Y. Eliashberg.
\newblock Classification of overtwisted contact structures on $3$-manifolds.
\newblock {\em Invent. Math.} 98:3, 623--637, 1989.

\bibitem{Ghrist00_2}
J.~Etnyre and R.~Ghrist.
\newblock Contact topology and hydrodynamics. {I}. {B}eltrami fields and the
  {S}eifert conjecture.
\newblock {\em Nonlinearity}, 13(2):441--458, 2000.

\bibitem{Giroux01}
E.~Giroux.
\newblock Structures de contact sur les vari\'et\'es fibr\'ees en cercles
  audessus d'une surface.
\newblock {\em Comment. Math. Helv.}, 76(2):218--262, 2001.

\bibitem{Komendarczyk_nodal}
R.~Komendarczyk.
\newblock On the contact geometry of nodal sets, {\tt math.dg/0402070}.
\newblock To appear, {\em  Trans. of Amer. Math. Soc.}, 2004.

\bibitem{Nicolaescu98}
L.~Nicolaescu.
\newblock Adiabatic limits of the {S}eiberg-{W}itten equations on {S}eifert
  manifolds.
\newblock {\em Comm. Anal. Geom.}, 6(2):331--392, 1998.

\bibitem{Rosenberg97}
S.~Rosenberg.
\newblock {\em The {L}aplacian on a {R}iemannian manifold}, volume~31 of {\em
  London Mathematical Society Student Texts}.
\newblock Cambridge University Press, Cambridge, 1997.
\newblock An introduction to analysis on manifolds.

\bibitem{Uhlenbeck76}
K.~Uhlenbeck.
\newblock Generic properties of eigenfunctions.
\newblock {\em Amer. J. Math.}, 98(4):1059--1078, 1976.

\bibitem{Yoshida90}
Z.~Yoshida and Y.~Giga.
\newblock Remarks on spectra of operator rot.
\newblock {\em Math. Z.}, 204(2):235--245, 1990.

\end{thebibliography}
%\input beltrami.bbl

\end{document}

math0411319.extracted_bib
\begin{thebibliography}{10}

\bibitem{Khesin98}
V.~Arnold and B.~Khesin.
\newblock {\em Topological Methods in Hydrodynamics}, volume 125 of {\em
  Applied Mathematical Sciences}.
\newblock Springer-Verlag, New York, 1998.

\bibitem{Benn}
D. Bennequin.
\newblock Entrelacements et \'{e}quations de Pfaff.
\newblock {\em Ast\'{e}risque}, 87--161, Soc. Math. France, Paris, 1983.

\bibitem{Blair02}
D.~Blair.
\newblock {\em Riemannian geometry of contact and symplectic manifolds}, volume
  203 of {\em Progress in Mathematics}.
\newblock Birkh\"auser Boston Inc., Boston, MA, 2002.

\bibitem{Chandrasekhar57}
S.~Chandrasekhar and P.~Kendall.
\newblock On force-free magnetic fields.
\newblock {\em Astrophys. J.}, 126:457--460, 1957.

\bibitem{Chicone}
C. Chicone.
\newblock The topology of stationary curl parallel solutions of
Euler's equations.
\newblock {\em Israel J. Math.} 39:1-2, 161--166, 1981.


\bibitem{Hamilton85}
S.~S. Chern and R.~S. Hamilton.
\newblock On {R}iemannian metrics adapted to three-dimensional contact
  manifolds.
\newblock In {\em Workshop Bonn 1984 (Bonn, 1984)}, volume 1111 of {\em Lecture
  Notes in Math.}, pages 279--308. Springer, Berlin, 1985.

\bibitem{Eli}
Y. Eliashberg.
\newblock Classification of overtwisted contact structures on $3$-manifolds.
\newblock {\em Invent. Math.} 98:3, 623--637, 1989.

\bibitem{Ghrist00_2}
J.~Etnyre and R.~Ghrist.
\newblock Contact topology and hydrodynamics. {I}. {B}eltrami fields and the
  {S}eifert conjecture.
\newblock {\em Nonlinearity}, 13(2):441--458, 2000.

\bibitem{Giroux01}
E.~Giroux.
\newblock Structures de contact sur les vari\'et\'es fibr\'ees en cercles
  audessus d'une surface.
\newblock {\em Comment. Math. Helv.}, 76(2):218--262, 2001.

\bibitem{Komendarczyk_nodal}
R.~Komendarczyk.
\newblock On the contact geometry of nodal sets, {\tt math.dg/0402070}.
\newblock To appear, {\em  Trans. of Amer. Math. Soc.}, 2004.

\bibitem{Nicolaescu98}
L.~Nicolaescu.
\newblock Adiabatic limits of the {S}eiberg-{W}itten equations on {S}eifert
  manifolds.
\newblock {\em Comm. Anal. Geom.}, 6(2):331--392, 1998.

\bibitem{Rosenberg97}
S.~Rosenberg.
\newblock {\em The {L}aplacian on a {R}iemannian manifold}, volume~31 of {\em
  London Mathematical Society Student Texts}.
\newblock Cambridge University Press, Cambridge, 1997.
\newblock An introduction to analysis on manifolds.

\bibitem{Uhlenbeck76}
K.~Uhlenbeck.
\newblock Generic properties of eigenfunctions.
\newblock {\em Amer. J. Math.}, 98(4):1059--1078, 1976.

\bibitem{Yoshida90}
Z.~Yoshida and Y.~Giga.
\newblock Remarks on spectra of operator rot.
\newblock {\em Math. Z.}, 204(2):235--245, 1990.

\end{thebibliography}