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\begin{document}
\title[Lagrangian graphs and minimizing measures]{Lagrangian graphs,
minimizing measures and Ma\~n\'e's critical values}
\author[G. Contreras]{Gonzalo Contreras}\thanks{Gonzalo Contreras was
partially supported by CNPq-Brasil.}
\address{Matem\'atica, PUC-Rio\\
R. Marqu\^es de S\~ao Vicente, 225\\
22453-900 Rio de Janeiro,
Brasil.}
\email{gonzalo@@mat.puc-rio.br}
\author[R. Iturriaga]{Renato Iturriaga}\thanks{Reanto Iturriaga was partially
supported by a CONACYT-M\'exico grant $\#$ 0324P-E}
\address{CIMAT \\
A.P. 402, 3600 \\
Guanajuato. Gto. \\
M\'exico.}
\email{renato@@fractal.cimat.mx}
\author[G. P. Paternain]{Gabriel P. Paternain}\thanks{Gabriel P.
and Miguel Paternain were partially supported by
a CONICYT-Uruguay grant $\#$ 301}
\address{Centro de Matem\'atica\\
Facultad de Ciencias\\
Eduardo Acevedo 1139\\
Montevideo CP 11200\\
Uruguay}
\email {gabriel@@cmat.edu.uy}
\author[M. Paternain]{Miguel Paternain}
\address{Centro de Matem\'atica\\
Facultad de Ciencias\\
Eduardo Acevedo 1139\\
Montevideo CP 11200\\
Uruguay}
\email{miguel@@cmat.edu.uy}
\date{1997}
%\maketitle
\begin{abstract} Let $L$ be a convex superlinear Lagrangian
on a closed connected manifold $M$. We consider critical
values of Lagrangians as defined by R. Ma\~n\'e in \cite{Ma}.
We show that the critical value of the lift of $L$ to a covering of $M$ equals the infimum
of the values of $k$ such that the energy level $k$
bounds an exact Lagrangian graph in the cotangent bundle
of the covering. As a consequence we show that up
to reparametrization, the dynamics of the Euler-Lagrange flow
of $L$ on an energy level that contains
minimizing measures with
nonzero homology can be reduced to Finsler metrics.
We also show that if the Euler-Lagrange flow of $L$ on the energy
level $k$ is Anosov, then $k$ must be strictly bigger than
the critical value $c_{u}(L)$
of the lift of $L$ to the universal covering
of $M$.
It follows that given $k$ denote the integral of $\omega$ on any closed curve in the homology class $\rho$.
If $\mu\in {\cal M}(L)$, its {\it homology} is defined as the unique $\rho(\mu)\in H_{1}(M,\re)$ such that
\[<\omega,\rho(\mu)>=\int\omega\,d\mu,\]
for all closed one-forms on $M$.
The integral on the right-hand side is with respect to $\mu$ with $\omega$ considered as a function $\omega:TM\rightarrow \re$.
The function $\rho:{\cal M}(L)\rightarrow H_{1}(M,\re)$ is surjective \cite{M}. The homology of an invariant measure is essentially
Schwartzman's asymptotic cycle \cite{S}.
The {\it action} of $\mu\in {\cal M}(L)$ is defined by
\[A_{L}(\mu)=\int L\,d\mu.\]
Finally we define the function $\beta:H_{1}(M,\re)\rightarrow \re$ by
\[\beta(\gamma)=\inf\{A_{L}(\mu):\;\rho(\mu)=\gamma\}.\]
The function $\beta$ is {\it convex} and {\it superlinear} and the infimum can be shown to be a {\it minimum} \cite{M} and the measures at which the minimum is attained are called {\it minimizing measures}.
In other words, $\mu\in {\cal M}(L)$ is a minimizing measure iff
\[\beta(\rho(\mu))=A_{L}(\mu).\]
Let us recall how the convex dual $\alpha:H^{1}(M,\re)\rightarrow \re$ of $\beta$ is defined.
Since $\beta$ is convex and superlinear we can set
\[\alpha([\omega])=\max\{<\omega,\gamma>-\beta(\gamma):\;\gamma\in H_{1}(M,\re)\},\]
where $\omega$ is any closed one-form whose cohomology class is $[\omega]$.
The function $\alpha$ is also convex and superlinear.
It is not difficult to see that \cite{M}
\begin{equation}
\alpha([\omega])=-\min\left\{\int (L-\omega)\,d\mu :\;\mu\in{\cal M}(L)\right\}. \label{coho}
\end{equation}
Let ${\cal M}^{\omega}(L)$ denote the set of all minimizing measures $\mu$
such that $[\omega]$ is the slope
of a supporting hyperplane through
$(\rho(\mu),\beta(\rho(\mu)))$.
M.J. Dias Carneiro proved \cite{DC} that
if $\mu$ is a minimizing measure in ${\cal M}^{\omega}(L)$,
then its support is contained in the energy level $k$ with $k=\alpha([\omega])$.
The purpose of the present paper is to present a new geometric way
of describing Mather's $\alpha$ function and Ma\~n\'e's notion of critical value of a Lagrangian \cite{Ma}. This approach has many interesting
applications that we describe below. Recall
that a smooth one form $\omega$ is a section of the bundle $T^{*}M\mapsto M$.
Let $G_{\omega}\subset T^{*}M$ be the graph of $\omega$.
It is well known that $G_{\omega}$ is a Lagrangian submanifold of $T^{*}M$ if and only if $\omega$ is closed. When $\omega$ is exact we shall say
that $G_{\omega}$ is an {\it exact Lagrangian graph}.
Let us define a function $h:H^{1}(M,\re)\to \re$ by
\[\quad\hbox{\rm $h(q)=\inf\{k\in\re$ : $H^{-1}(-\infty,k)$ contains a
Lagrangian graph $G_{\omega}$ with $[\omega]= q$}\}.\]
\begin{Theorem}$\alpha\equiv h$. \label{Main}
\end{Theorem}
Theorem \ref{Main} is a consequence of a more general theorem to be stated
below and whose proof is based on the notion of critical values introduced
by Ma\~n\'e in \cite{Ma} and that we now recall.
The action of the Lagrangian $L$ on
an absolutely continuous curve $u:[a,b]\rightarrow M$ is defined by
\[A_{L}(u)=\int_{a}^{b}L(u(t),\dot{u}(t))\,dt.\]
Given two points, $x_{1}$ and $x_{2}$ in $M$, denote by ${\cal C}(x_{1},x_{2})$ the set of absolutely continuous curves $u:[0,T]\rightarrow M$, with $u(0)=x_{1}$ and $u(T)=x_{2}$.
For each $k\in \re$ we define the {\it action potential} $\Phi_{k}:M\times M\rightarrow \re$ by
\[\Phi_{k}(x_{1},x_{2})=\inf\{A_{L+k}(u):\;u\in {\cal C}(x_{1},x_{2})\}.\]
Ma\~n\'e showed \cite{Ma, CDI} that there exists $c(L)\in \re$ such that
\begin{itemize}
\item if $k-\infty$ for all $x_{1}$ and $x_{2}$ and $\Phi_{k}$ is a Lipschitz function;
\item if $k\geq c(L)$, then
\[\p(x_{1},x_{3})\leq \p(x_{1},x_{2})+\p(x_{2},x_{3}),\]
for all $x_{1}$, $x_{2}$ and $x_{3}$ and
\[\p(x_{1},x_{2})+\p(x_{2},x_{1})\geq 0,\]
for all $x_{1}$ and $x_{2}$;
\item if $k>c(L)$, then for $x_{1}\neq x_{2}$ we have
\[\p(x_{1},x_{2})+\p(x_{2},x_{1})> 0.\]
\end{itemize}
Observe that in general the action potential $\p$ is {\bf not} symmetric, however defining $d_{k}:M\times M\rightarrow \re$ by
\[d_{k}(x_{1},x_{2})=\p(x_{1},x_{2})+\p(x_{2},x_{1}),\]
the properties above say that $d_{k}$ is a metric for $k>c(L)$ and a pseudometric for $k=c(L)$.
The number $c(L)$ is called the {\it critical value} of $L$.
It is important for our purposes to indicate that the results above also hold for
coverings of $M$, i.e. suppose $\widehat{M}$ is a covering of $M$ with
covering projection $p$.
Take the lift of the Lagrangian $L$ to $\widehat{M}$ which is given by
\[\widehat{L}(x,v)=L(p(x),dp(v)).\]
Then we define for each $k\in \re$ the action potential $\q$ just as above
and the results hold for $\widehat{L}$. Thus we have a critical value for $\widehat{L}$.
Moreover, if $M_{1}$ and $M_{2}$ are coverings of $M$ such that $M_{1}$
covers $M_{2}$, then
\begin{equation}
c(L_{1})\leq c(L_{2}), \label{cubre}
\end{equation}
where $L_{1}$ and $L_{2}$ denote the lifts of the Lagrangian $L$ to $M_{1}$
and $M_{2}$ respectively.
Note that if $M_{1}$ is a {\it finite} covering of $M_{2}$ then
$c(L_{1})= c(L_{2}). $
Among all possible coverings of $M$ there are two distinguished ones; the
universal covering which we shall denote by $\widetilde{M}$, and the abelian
covering which we shall denote by $\overline{M}$.
The latter is defined as the covering of $M$ whose fundamental group
is the kernel of the Hurewicz homomorphism
$\pi_{1}(M)\mapsto H_{1}(M,\re)$.
When $\pi_{1}(M)$ is abelian, $\widetilde{M}$ is a finite covering of
$\overline{M}$.
The universal covering of $M$ gives rise to the critical value
\[c_{u}(L)\df c(\widetilde{L}),\]
and the abelian covering of $M$ gives rise to the critical value
\[c_{a}(L)\df c(\overline{L}).\]
>From inequality (\ref{cubre}) it follows that
\[ c_{u}(L)\leq c_{a}(L),\]
but in general the inequality may be strict as it was shown in \cite{PP3}.
Ma\~n\'e \cite{Ma,CDI} established a connection between the critical values
of a Lagrangian and $\alpha$, the convex dual of Mather's $\beta$ function.
He showed that
\begin{equation}
c(L)=-\min\left\{\int L\,d\mu:\,\mu\in{\cal M}(L)\right\}, \label{ergodic}
\end{equation}
and therefore combining (\ref{coho}) and (\ref{ergodic}) we obtain the remarkable equality
\begin{equation}
c(L-\omega)=\alpha([\omega]), \label{notable}
\end{equation}
for any closed one-form $\omega$ whose cohomology class is $[\omega]$.
Finally, Ma\~n\'e defined the {\it strict critical value} of $L$ as
\[c_{0}(L)\df \min\{c(L-\omega):\;[\omega]\in H^{1}(M,\re)\}=-\beta(0).\]
It was shown in \cite{PP3} that the strict critical value of $L$
equals the critical value of the lift of $L$ to the abelian covering of $M$,
that is, $c_{a}(L)=c_{0}(L)$.
For a given covering let us define $g(\widehat{L})$ as follows:
\[\quad\hbox{\rm $g(\widehat{L})=\inf\{k\in\re:$ $\widehat{H}^{-1}(-\infty,k)$ contains an exact
Lagrangian graph}\},\]
where $\widehat{H}$ is the Hamiltonian associated with $\widehat{L}$.
In Section 2 we shall show
\begin{Theorem} $c(\widehat{L})=g(\widehat{L})$. \label{Main2}
\end{Theorem}
Let us explain why Theorem \ref{Main} follows from Theorem \ref{Main2}.
Since $\alpha([\omega])=c(L-\omega)$, it sufficies to show
that $g(L-\omega)=h([\omega])$. All the closed one forms in the class
$[\omega]$ are given by $\omega+df$ where $\omega$ is fixed and $f$
ranges among all smooth functions.
Observe that the Hamiltonian
associated with $L-\omega$ is $H_{\omega}(x,p)\df H(x,p+\omega_{x})$
and therefore $H^{-1}(-\infty,k)$ contains the Lagrangian
graph $G_{\omega+df}$ if and only if $H_{\omega}^{-1}(-\infty,k)$ contains
the exact Lagrangian graph $G_{df}$ and thus $g(L-\omega)=h([\omega])$.
Note that saying that $\widehat{H}^{-1}(-\infty,k)$ contains
an exact Lagrangian graph is equivalent to saying that there exist strict
smooth subsolutions $f:\widehat{M}\to \re$ of the Hamilton-Jacobi
equation $\widehat{H}(x,d_{x}f)=k$.
In fact when $k=c(L)$ a recent result of A. Fathi \cite{Fa1} says that there
exist Lipschitz viscosity solutions of the Hamilton-Jacobi equation
$H(x,d_{x}f)=c(L)$ and that $c(L)$ is the only real number for which
this happens.
Theorem \ref{Main2} has the following interesting corollary whose proof
will also be given in Section 2.
\begin{Corollary} If $k>c(\widehat{L})$, then it is possible
to see the dynamics of $\phi_{t}|_{ \widehat{E}^{-1}(k)}$ as the reparametrization of the geodesic flow
on the unit tangent bundle of an appropriatedly chosen Finsler metric
on $\widehat{M}$. \label{cor1}
\end{Corollary}
In particular, if we take $k>c_{0}(L)$ then it is possible
to see the dynamics of $\phi_{t}|_{E^{-1}(k)}$ as the
reparametrization of the geodesic flow
on the unit tangent bundle of an appropriatedly chosen Finsler metric
on $M$. Simply apply the previous corollary to the Lagrangian $L-\omega$
where $\omega$ is a closed one form such that $c_{0}(L)=c(L-\omega)$.
Finally, let us describe another application of Theorem \ref{Main2}.
Let $\pi:TM\to M$ denote the canonical projection and,
if $(x,v)\in TM$, let $V(x,v)$ denote the vertical fibre at $(x,v)$ defined
as usual as the kernel of $d\pi_{(x,v)}:T_{(x,v)}TM\to T_{x}M$.
Let us set
\[e=\max_{x\in M}E(x,0)=-\min_{x\in M}L(x,0).\]
Note that the energy level $E^{-1}(k)$ projects {\it onto} the manifold
$M$ if and only if $k\geq e$ and for any $k>e$, the energy level $E^{-1}(k)$ is a smooth closed hypersurface of $TM$ that intersects each tangent space $T_{x}M$ in a sphere containing the origin in its interior.
It is quite easy to check that the inequality $e\leq c_{u}(L)$ always holds,
but in general the inequality may be strict (cf. \cite{PP3}).
An {\it Anosov energy level} is a regular energy level on
which the flow $\phi_{t}$ is an Anosov flow. G.P. Paternain
and M. Paternain showed in \cite{PP} that Anosov energy levels are free
of conjugate points and that they must project onto the whole manifold thus generalizing a well known result
of Klingenberg \cite{K} for geodesic flows (cf. also \cite{M-1}).
Conjugate points, means, as usual, pair of
points $(x_{1},v_{1})\neq (x_{2},v_{2})=\phi_{t}(x_{1},v_{1})$ such
that $d\phi_{t}(V(x_{1},v_{1}))$ intersects $V(x_{2},v_{2})$
non-trivially.
Moreover in \cite{PP3} they showed that if there exists $k$ such that for all $k'\geq k$, the energy level $k'$ is Anosov, then $k> c_{u}(L).$
In Section 3 we shall complete these results by showing:
\begin{Theorem}If the energy level $k$ is Anosov, then
\[k> c_{u}(L).\] \label{anosov}
\end{Theorem}
In \cite{PP3}, G.P. Paternain and M. Paternain gave examples of Anosov energy
levels $k$ with $k0$ there exists a smooth function $\psi:M\to \re$ with $|\psi|_{C^{2}}<\varepsilon$ and such that the energy level
$k$ of $L+\psi$ possesses conjugate points. \label{cor2}
\end{Corollary}
We remark that if $k$ is a regular value of the energy such that $k} and
Proposition \ref{<} below.
\begin{Lemma}If there exists a $C^{1}$ function $f:\widehat{M}\to\re$
such that $\widehat{H}(df)c(\widehat{L})$. \label{>}
\end{Lemma}
\PROOF recall that
\[\widehat{H}(x,p)=\max_{v\in T_{x}\widehat{M}}\{p(v)-\widehat{L}(x,v)\}.\]
Since $\widehat{H}(df)0$ we have
\[\int_{0}^{T}(L(u,\dot{u})+k)\,dt=\int_{0}^{T}(L(u,\dot{u})+k+d_{u}f(\dot{u}))\,dt>0,\]
and thus $k>c(\hat{L})$.
\qed
\begin{Lemma} Let $k\geq c(\widehat{L})$.
If $f:\widehat{M}\to\re$ is differentiable at $x\in \widehat{M}$ and satisfies
$$
f(y)-f(x)\le\q (x,y)
$$
for all $y$ in a neighbourhood of $x$, then $H\bigl(d_{x}f\bigr)\le k$.
\end{Lemma}
\PROOF let $u(t)$ be a differentiable curve on $\widehat{M}$ with
$(u(0),\dot{u}(0))=(x,v)$. Then
$$
\limsup_{t\to 0^+} \frac{f(u(t))-f(x)}{t}
\le \liminf_{t\to 0^+} \frac{1}{t}\int_0^t\bigl[ \widehat{L}(u,\dot{u})+k\bigr]\, ds.
$$
Hence $d_{x}f(v) \le \widehat{L}(x,v)+k$ for all $v\in T_x\widehat{M}$ and thus
\[\widehat{H}(x,d_{x}f)=\max_{v\in T_{x}\widehat{M}}\{d_{x}f(v)-\widehat{L}(x,v)\}\leq k.\]
\qed
\bigskip
\begin{Proposition}
For any $k>c(\widehat{L})$ there
exists $f\in C^\infty(\widehat{M},\re)$ such that
$\widehat{H}(df) 0$ be such that
\begin{itemize}
\item[(a)] The $3\e$-neighbourhood of $M$ in $\re^N$ is
contained in $U$.
\item[(b)] If $x\in M$, $(y,p)\in T^*\re^N= \re^{2N}$,
${\Bbb H}(y,p)\le c$ and $d_{\re^N}(x,y)<\e$
then ${\Bbb H}(x,p)From the choice of $\e>0$ we have that
${\Bbb H}(x,d_{y}\ov{g})< k$ for almost every
$y\in\text{supp}K(x,\cdot)$ and $x\in M$. Since $K(x,y)\,dy $ is a
probability measure, by Jensen's inequality
$$
H(d_{x}f)
\le {\Bbb H}( d_{x}f)
\le \int_{\re^N} {\Bbb H}(x, d_{y}\ov{g})\, K(x,y)\; dy
< k\,.
$$
for all $x\in M$.
Now, suppose that $\widehat{M}$ is a covering of $M$. Fix $q\in \widehat{M}$
and set $g(x)=\widehat{\Phi}_{c(\widehat{L})}(x,q)$
We can regularize our function $g$ similarly as we
now explain.
For $\tx\in\widehat{M}$ let $x$ be the projection of $\tx$
to $M$ and let $\mu_{x}$ be
the Borel probability measure on $M$ defined by
\[\int_{M}\vr\,d\mu_{x}=\int_{\re^{n}}(\vr\circ\rho)(y)K(x,y)\,dy,\]
for any continuous function $\vr:M\to \re$.
Then the support of $\mu_{x}$ satisfies
\[\text{supp}(\mu_{x})\subset \{y\in M\,:\,d_{M}(x,y)<\varepsilon\}.\]
Let $\widehat{\mu}_{\tx}$ be the lift of $\mu_{x}$ with
$\text{supp}(\widehat{\mu}_{\tx})\subset \{\widehat{y}\in \widehat{M}\,:\,d_{\widehat{M}}(\tx,\widehat{y})<\varepsilon\}$.
Then we have
\[\frac{d}{d\tx}\int_{\widehat{M}}\vr\,d\widehat{\mu}_{\tx}=
\int_{\widehat{M}}d_{\widehat{y}}\vr\, d\widehat{\mu}_{\tx}(\widehat{y}),\]
for any weakly differentiable fucntion $\vr:\widehat{M}\to\re$.
The same arguments as above show that
\[f(\tx)=\int_{\widehat{M}}g(\widehat{y})\,d\widehat{\mu}_{\tx}(\widehat{y}),\]
satisfies $H(d_{\tx}f)c(\widehat{L})$, then $\widehat{H}^{-1}(-\infty,k)$
contains an exact Lagrangian graph. This means that there exists
a smooth function $f:\widehat{M}\to \re$ such that $\widehat{H}(x,d_{x}f)} and the fact that there exists $\varepsilon>0$
such that for all $k'\in (k-\varepsilon,k+\varepsilon)$ the
energy level $k'$ is Anosov.
\qed
Let us prove now Corollary \ref{cor2}.
%We shall prove first that if $\psi_{n}$ converges to the zero function in the
%$C^{2}$ topology then
%\begin{equation}
%\liminf_{n\to\infty}c(\widehat{L}+\widehat{\psi}_{n})\geq c(\widehat{L}).
%\label{semi}
%\end{equation}
%Take $k0$ such that for every $\psi$ with
$|\psi|_{C^2} <\epsilon $, the energy level $k$ of $L+\psi$ has no conjugate
points. The main result in \cite{CIS} says that in this case
the energy level $k$ of $L$ must be Anosov thus contradicting Theorem \ref{anosov}.
\qed
\begin{Proposition}If $k$ is a regular value of the energy such
that $k