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Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \newcommand{\exercise}{% {\bf \green EXERCISE:\ }} \newcommand{\proof}{% {\bf \green Proof:\ }} \newcommand{\pstep}{% {\bf \green Proof. Step 1:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Symplectic geometry\\[15mm] \small lecture 7: Ekeland-Hofer theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE, room 306, 16:20, \\[2mm] September 25, 2021 } \end{center} \newpage {\bf \blue Gromov capacity (reminder)} \definition {\bf \green An (open) symplectic embedding} is an open embedding of symplectic manifolds, symplectimorphic to its image. \definition Let $(M,\omega)$ be a symplectic manifold, and $r$ a supremum of radii of all symplectic balls of the same dimension, admitting a symplectic embedding to $M$. The number $\capa(M,\omega):=\pi r^2$ is called {\bf \blue Gromov symplectic capacity} of $M$. \theorem {\bf \blue (Ekeland-Hofer)}\\ Let $\phi$ be an oriented diffeomorphism of symplectic manifold. Then {\bf \red $\phi$ is a symplectomorphism if and only if $\phi$ preserves the symplectic capacity of all open subsets.} \proof Later today. \endproof \newpage {\bf \blue Ekeland-Hofer theorem (reminder)} \theorem {\bf \blue (Ekeland-Hofer)}\\ Let $\phi$ be an oriented diffeomorphism of symplectic manifold. Then {\bf \red $\phi$ is a symplectomorphism if and only if $\phi$ preserves the symplectic capacity of all open subsets.} Today, the Ekeland-Hofer theorem {\bf \green will be deduced from its linear version.} \theorem {\bf \blue Ekeland-Hofer, the linear version}\\ Let $(V=\R^{2n}, \omega= \sum_i dp_i \wedge dq_i)$ be a symplectic vector space, and $\phi:\; V \arrow V$ an oriented linear map which preserves the Gromov capacity of all ellipsoids. {\bf \red Then $\phi$ is a symplectomorphism.} \proof Next lecture. \endproof \newpage {\bf \blue Hausdorff metric (reminder)} \definition Let $Z\subset M$ be a metric space. {\bf \blue An $\epsilon$-neighbourhood} $Z(\epsilon)$ of $Z$ is a union of all $\epsilon$-balls centered in $Z$. {\bf \blue Hausdorff metric} $d_H$ on closed subsets $M$ is defined as follows: $d_H(X,Y)$ is infimum of all $\epsilon$ such that $Y\subset X(\epsilon)$ and $X\subset Y(\epsilon)$. {\bf \green Properties of Hausdorff metric:} 1. Let $M$ be a metrizable topological space, and ${\cal C}$ the set of compact subsets of $M$. Then the topology induced by the Hausdorff metric on ${\cal C}$ is independent from the choice of the metric on $M$. {\bf \purple (Prove it!)} 2. Let $\phi_i:\; M \arrow N$ be a sequence of continuous maps, and $\Gamma_{\phi_i}\subset M\times N$ their graphs. Suppose that $M, N$ are compact. Then the sequence $\{\phi_i\}$ converges to $\phi:\; M \arrow N$ (in the compact-open topology) {\bf \purple if and only if $\Gamma_{\phi_i}$ converges to $\Gamma_\phi$ in the Hausdorff topology.} \remark Further on, we will consider ``the Hausdorff metric'' on the set of open subsets of $M$ with compact closure. {\bf \red This is in fact a pseudometric:} two open subsets $U, V$ with the same closure satisfy $d_H(U, V)=0$. \newpage {\bf \blue Hausdorff distance and boundary} %\lemma %Let $E\subset \R^n$ be a convex set. %{\bf \red Then $E(\epsilon)$ is convex. } % %\proof %Given an interval $a_1, a_2\in E(\epsilon)$, %take $b_1, b_2 \in E$ such that $d(a_i, b_i)< \epsilon$. %Then the distance between any point of %$[a_1, a_2]$ and the interval $[b_1, b_2]\subset E$ %is less than $\epsilon$, hence %$[a_1, a_2]\subset E(\epsilon)$. %\endproof \lemma Let $E_1, E_2$ be bounded open convex subsets in $\R^n$ and $U_i := \R^n \backslash E_i$. Then $d_H(E_1, E_2) \geq d_H(U_1, U_2)$. \proof Suppose that $d_H(U_1, U_2)\geq \epsilon$. Let $x\in U_1$ such that $x\notin U_2(\epsilon)$. This means that $d(x, U_2) \geq \epsilon$, hence $B_\epsilon(x) \supset E_2$. By Hahn-Banach separation theorem, there exists a hyperplane $H$ passing through $x$ such that all $E_1$ lies to one side of this hyperplane. Since the distance from the farthest point of $B_\epsilon(x)$ to this hyperplane is $\epsilon$, and $B_\epsilon(x)\subset E_2$, this implies that $d_H(E_1, E_2)\geq \epsilon$. \endproof \lemma Let $E_1, E_2$ be bounded open convex subsets in $\R^n$ and $\6(E_1):= \bar E_i \backslash E_i$ denote {\bf \blue the boundary} of $E_i$. {\bf \red Then $d_H(E_1, E_2) \geq d_H(\6 E_1, \6 E_2)$.} \proof Whenever $d_H(E_1, E_2)< \epsilon$, we also have $d_H(U_1, U_2)< \epsilon$, where $U_i = \R^n \backslash E_i$, hence every point $x$ on a boundary of $E_1$ satisfies $x\in E_2(\epsilon) \cap U_2(\epsilon) =\6 E_2 (\epsilon)$. \endproof \newpage {\bf \blue Hausdorff distance between convex sets and homothety} {\bf \green Claim 1:} Let $E\subset \R^n$ be a convex set containing 0, and $\epsilon >0$. Define $\lambda U:= \{x\in \R^{2n}\ \ | \ \ \lambda^{-1}x \in U\}$. {\bf \red Then for each $\epsilon>0$ there exists $\delta>0$ such that any convex $E_1$ satisfying $d_H(E, E_1) < \epsilon$ also satisfies $(1-\delta) E\subset E_1\subset (1+\delta) E$.} \pstep Define $u(\delta):\;\R^{>0} \arrow \R^{>0}$ as \[ u(\delta):= \inf_{x\in \6 E}\min [d(x, \6 (1-\delta) E), d(x, \6 (1+\delta) E)]. \] Since $\6 E$ is compact, and $d(x, \6 (1-\delta) E)$ is 1-Lipschitz as a function of $x$, the number $d(x, \6 (1-\delta) E)$ reaches its minumum somewhere on $\6 E$, and $u(\delta)$ is positive. An infimum of 1-Lipschitz functions is 1-Lipschitz, hence $u$ is 1-Lipschitz. {\bf \purple Therefore, there exists $\delta$ such that $u(\delta) < \epsilon$.} {\bf \green Step 2:} The inequality $d_H(E, E_1) < \epsilon $ implies $\6E_1 \subset \6E(\epsilon)$, by the previous lemma. Step 1 implies that {\bf \purple anything which lies in an $\epsilon$-neighbourhood of $\6E$ belongs to the segment bounded by $\6 (1-\delta)E$ and $\6(1-\delta) E)$.} This implies that $(1-\delta) E\subset \6E_1\subset (1+\delta) E$. Passing to convex hulls, we obtain $(1-\delta) E\subset E_1\subset (1+\delta) E$. \endproof \newpage {\bf \blue Symplectic capacity and Hausdorff convergence (remider)} \claim Let $U_i\subset \R^{2n}$ be a Cauchy sequence of bounded open subsets, containing 0, and $U$ their limit in the Hausdorff metric. Assume that $U$ is convex. {\bf \red Then $\lim_i\capa_G(U_i)=\capa_G(U)$,} where $\capa_G(U)$ denotes the Gromov capacity. \proof Let $\lambda U:= \{x\in \R^{2n}\ \ | \ \ \lambda^{-1}x \in U\}$. By Claim 1, for any $\epsilon >0$, almost all elements of the sequence $U_i$ contain $(1-\epsilon)U$ and are contained in $(1+\epsilon)U$. In this situation, \[ \sqrt{1-\epsilon}\capa_G(U)\leq \capa_G(U_i)\leq \sqrt{1+\epsilon}\capa_G(U). \] \endproof Now we can deduce Ekeland-Hofer theorem from its linear version. \theorem {\bf \blue Ekeland-Hofer, the linear version}\\ Let $(V=\R^{2n}, \omega= \sum_i dp_i \wedge dq_i)$ be a symplectic vector space, and $\phi:\; V \arrow V$ an oriented linear map which preserves the Gromov capacity of all ellipsoids. {\bf \red Then $\phi$ is a symplectomorphism.} \newpage {\bf \blue Image of an ellipsoid} Recall that {\bf \blue $C^n$-topology} on functions $\phi:\; \R^n \arrow \R^k$ is the topology of uniform convergence on compacts for the derivatives up to $n$-th. {\bf \green Claim 2:} Let $\phi_i:\; \R^n \arrow \R^n$ be a sequence of diffeomorphisms converging to identity in $C^2$-topology, and $E\subset \R^n$ an ellipsoid. {\bf \red Then for $i$ sufficiently big, all $\phi_i(E)$ are convex.} \proof Let $u:\; \R^n \arrow \R$ be a quadratic function $u(x_1, ..., x_n)= \sum x_i^2 a_i^2$ such that $E= u^{-1}([0,1[)$, and $u_i:= (\phi_i^{-1})(u)$. Then $\phi_i(E)= u^{-1}_i([0,1[)$. Clearly, the $C^2$-convergence of $\phi_i$ to identity implies the $C^2$-convergence of $u_i$ to $u$. The functions $u_i$ are convex for $i$ sufficiently big, because {\bf \purple the Hessian $\Hess(u_i-u)$ uniformly converges to zero, and $\Hess(u)$ is positive definite.} On the other hand, the preimage $u^{-1}_i([0,1[)$ is convex if $u_i$ is convex. \endproof \newpage {\bf \blue Ekeland-Hofer theorem deduced from its linear version (reminder)} \theorem {\bf \blue (Ekeland-Hofer)}\\ Let $\phi$ be an oriented diffeomorphism of symplectic manifold. Then {\bf \red $\phi$ is a symplectomorphism if and only if $\phi$ preserves the symplectic capacity of all open subsets.} \proof Locally, every symplectic manifold is symplectomorphic to a symplectic ball (Darboux). Therefore it would suffice to prove the following (weaker) form of this theorem. \theorem Let $(B,\omega)\stackrel \phi \arrow (\R^{2n},\omega)$ be an open embedding, mapping a symplectic ball to $\R^{2n}$ with the usual symplectic structure, mapping 0 to 0, and preserving the Gromov symplectic capacities of all convex subsets. {\bf \red Then $\phi$ is a symplectomorphism}. \pstep Let $\lambda>1$, and $\Gamma_\lambda:\; \R^n \arrow \R^n$ the homothety mapping $v$ to $\lambda v$. By conformal invariance of $\capa_G$, the diffeomorphism $\phi_\lambda:\; B \arrow \R^{2n}$, defined as $\phi_\lambda(v):=\Gamma_\lambda(\phi(\Gamma_\lambda^{-1}(v)))$ preserves the Gromov symplectic capacities. {\bf \purple If $\phi(x)= \sum_{i=1}^\infty P_i(x)$ is the Taylor decomposition for $\phi$, with $P_i$ homogeneous polynomials of degree $i$, one has $\phi_\lambda(x) = \sum_{i=1}^\infty \lambda^{i-1}P_i(x)$.} \newpage {\bf \blue Ekeland-Hofer theorem deduced from its linear version (2)} {\bf \green Step 1:} {\bf \purple Let $\phi_\lambda(v):=\Gamma_\lambda(\phi(\Gamma_\lambda^{-1}(v)))$.} If $\phi(x)= \sum_{i=1}^\infty P_i(x)$ is the Taylor decomposition for $\phi$, we have $\phi_\lambda(x) = \sum_{i=1}^\infty \lambda^{i-1}P_i(x)$. {\bf \green Step 2:} For any diffeomorphism $(B,\omega)\stackrel \phi \arrow (\R^{2n},\omega)$ and any ellipsoid $E\subset B$, {\bf \purple there exists $\lambda_0>0$ such that $\phi_\lambda(E)$ is convex for any $\lambda>\lambda_0$.} Indeed, the second derivative of $\phi_\lambda$ tends to 0 as $\lambda$ tends to infinity, hence for $\lambda$ sufficiently big this map maps $E$ to a convex set. {\bf \green Step 3:} In an open-compact topology, $\lim\limits_{\lambda\rightarrow \infty}\phi_\lambda$ is equal to the differential ${\goth D}:=D_0\phi$. For each ellipsoid $E\subset B$, we have ${\goth D}(E)=\lim\limits_{\lambda\rightarrow \infty}\phi_\lambda(E)$ (in the Hausdorff topology). For $\lambda$ sufficiently big, the set $\phi_\lambda(E)$ is convex by Claim 2, and on convex subsets, the function $\capa_G$ is continuous in the Hausdorff topology. {\bf \purple This gives \[ \capa_G({\goth D}(E))=\lim\limits_{\lambda\rightarrow \infty} \capa_G(\phi_\lambda(E))=\capa_G(E), \] that is, ${\goth D}$ preserves the symplectic capacity.} {\bf \green Step 4:} Using the linear Ekeland-Hofer, we obtain that ${\goth D}=D_0\phi$ is a symplectomorphism. Since the choice of 0 was arbitrary, the same argument proves that {\bf \purple the differential of $\phi$ preserves the symplectic form everywhere where it is defined.} \endproof \end{document}