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Step 1:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Symplectic geometry\\[15mm] \small lecture 8: Ekeland-Hofer theorem (linear version deduced from non-squeezing)} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE, room 306, 16:20, \\[2mm] September 29, 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.} Last lecture, 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 Later today. \endproof \newpage {\bf \blue Symplectic homeomorphisms} \corollary Let $\phi:\; B \arrow \R^{2n}$ be an embedding from a symplectic ball to $(\R^n, \sum_i dp_i \wedge dq_i)$ which is locally a diffeomorphism. Assume that $\phi$ preserves the symplectic capacity of every ellipsoid $E$ such that $\phi(E)$ is convex. {\bf \red Then $\phi$ is a symplectomorphism.} \endproof The following theorem is deduced from Ekeland-Hofer. \theorem {\bf \blue (Eliashberg-Gromov)}\\ Let $(M, \omega)$ be a symplectic manifold. Then {\bf \red the group $\Symp(M)$ of symplectomorphisms is closed in the group $\Diff(M)$ of diffeomorphisms} with the open-compact topology. \newpage {\bf \blue Symplectic homeomorphisms} \theorem {\bf \blue (Eliashberg-Gromov)}\\ Let $(M, \omega)$ be a symplectic manifold. Then {\bf \red the group $\Symp(M)$ of symplectomorphisms is closed in the group $\Diff(M)$ of diffeomorphisms} with the open-compact topology. \proof It would suffice to prove a weaker statement: let $\phi_i:\; B \arrow \R^{2n}$ be a sequence of symplectic embeddings, converging in $C^0$ to a smooth embedding $\phi:\; B \arrow \R^{2n}$. Then $\phi$ is a symplectomorphism. Let $U\subset \phi(B)$ be a convex set, such that $E:=\phi^{-1}(U)$ is convex, and $E_i:= \phi_i^{-1}(U)$. Then $\lim_i(d_h(E_i, E))=0$. Since $\phi_i$ converges to $\phi$ in $C^0$, we have $\lim_i d_H(\6E, \6E_i)=0$. Therefore, for any $\epsilon >0$, almost all $E_i$ satisfy $(1-\epsilon) E \subset \6 E_i \subset (1+\epsilon) E$. This implies that the symplectic capacity of $E_i$ converges to $\capa_G(E)$. On the other hand, $\capa_G(E_i) = \capa_G(U)$, hence $\capa_G(U)= \lim_i \capa_G(E_i)= \capa_G(E)$, and $\phi_i$ preserves capacities. \endproof \definition {\bf \blue Symplectic homeomorphism} is the closure of the group of symplectic diffeomorphisms in the group of homeomorphisms, with respect to the $C^0$-topology. \newpage {\bf \blue Symplectic ellipsoids} {\bf \green Claim 3:} Let $V=\R^{2n}$, $g$ a positive definite bilinear symmetric form on $\R^n$, and $\omega$ a non-degenerate antisymmetric 2-form. Then there exists a $g$-orthonormal basis $v_i$ in $V$ and a set of positive real numbers $\alpha_1, ..., \alpha_n$, independent from the choice of a basis, such that \[ \omega=\begin{pmatrix} \omega=\begin{matrix}0 & \alpha_1\\ -\alpha_1 & 0\end{matrix} & & 0 \\ & \ddots & \\ 0 & & \begin{matrix}0 & \alpha_n \\ -\alpha_n & 0\end{matrix} \end{pmatrix} \ \ \ \ (*) \] {\bf \red In other words, $\omega = \sum_i \alpha_i x_i \wedge y_i$, where $x_1, y_1, x_2, y_2, ...\in V^*$ is the dual basis, and $\alpha_1 \geq \alpha_2 \geq ...\geq \alpha_n>0$ positive real numbers.} \proof Let $A:= g^{-1}\omega$, that is, $g(Ax,y)=\omega(x,y)$. {\bf \purple The operator $A$ is anti-symmetric: $g(Ax,y)= \omega(x,y)= - \omega(y, x)=-g(Ay,x)=-g(x,Ay)$.} An antisymmetric operator has a form (*) in some basis, which follows, for example, from the classification of rotations in the Euclidean space. \endproof \newpage {\bf \blue Symplectic ellipsoids (the second argument)} {\bf \green Another proof of Claim 3. Step 1:}\\ Since $g(Ax,y)= \omega(x,y)= - \omega(y, x)=-g(Ay,x)=-g(x,Ay)$, we have $g(A^2x,y)=g(x, A^2y)$: the operator $A^2$ is symmetric. Since $g(A^2x,x)=-g(Ax,Ax)$, {\bf \purple it is symmetric and negative definite.} {\bf \green Step 2:} Let $g_A(x, y) = - g(A^2x, y)$, and $\{x_i\}$ be the basis in which both $g$ and $g_A$ are diagonal, with $g(x_i, x_i)= 1$, $g_A(x_i, x_i)=b_i$. {\bf \purple Such a basis exists and is defined up to a map which is an isometry for both $g$ and $g_A$.} {\bf \green Step 3:} Since $g(Ax_i, Ax_j)= b_i\delta_{ij}$ and $g_A(Ax_i, Ax_j)= -b_i^2\delta_{ij}$, the vectors $b_i^{-\nicefrac 1 2}Ax_i$ are also orthogonal under $g$ and $g_A$. For all $i$, take for $x_{2i}$ the vector $b_{2i-1}^{-\nicefrac 1 2}Ax_{2i-1}$. In this basis, $A$ is written as \[ A=\begin{pmatrix} \omega=\begin{matrix}0 & \alpha_1\\ -\alpha_1 & 0\end{matrix} & & 0 \\ & \ddots & \\ 0 & & \begin{matrix}0 & \alpha_n \\ -\alpha_n & 0\end{matrix} \end{pmatrix}, \] where $\alpha_i =b_{2i-1}^{\nicefrac 1 2}$. \endproof \newpage {\bf \blue Normal form of symplectic ellipsoid} \definition Let $(V, \omega)$ be a symplectic vector space. {\bf \blue Symplectic basis} is a basis $\{x_1, y_1, x_2, y_2, ...\}$ such that $\omega=\sum x_i^*\wedge y_i^*$ where $x_i^*, y_i^*\in V^*$ is the dual basis. \definition {\bf \blue Normal form of an ellipsoid $E$ in a symplectic vector space $V$} is $E=\{v= \sum a_i x_i +b_i y_i \ \ | \ \ \sum_i(a_i^2+b_i^2)\alpha_i< 1\}$, where $\{x_i, y_i\}$ is a symplectic basis. \claim For any ellipsoid $E$ in a symplectic vector space, {\bf \red there exists a symplectic basis such that the ellipsoid is written in the normal form}, and the numbers $\alpha_i$ are determined uniquely by the ellipsoid and $\omega$. \proof Let $g$ be a bilinear symmetric form such that $E=\{v\in V\ \ |\ \ g(v,v)<1\}$, and $x_i', y_i'$ a $g$-orthonormal basis such that $\omega$ is written by the matrix (*). Then $x_i= x'_i \alpha_i^{-\nicefrac 1 2}, y_i =y'_i \alpha_i^{-\nicefrac 1 2}$ is a symplectic basis, and $E$ is written as $\{v= \sum a_i x_i +b_i y_i \ \ | \ \ \sum_i (a_i^2+b_i^2)\alpha_i< 1\}$. The numbers $\alpha_i$ are eigenvalues of $A= g\omega^{-1}$, hence they are determined uniquely. \endproof \newpage {\bf \blue Symplectic ellipsoids and capacity} \newcommand{\Cyl}{\operatorname{Cyl}} {\bf \green We compute the capacity of an ellipsoid using Gromov's non-squeezing theorem}. \claim Let $E$ be an ellipsoid in a symplectic space, written in the normal form $\{v= \sum a_i x_i +b_i y_i \ \ | \ \ \sum_i (a_i^2+b_i^2)\alpha_i< 1\}$, and $\alpha_1$ the smallest of the numbers $\alpha_i$. {\bf \red Then $\capa_G(E)=\pi \alpha_1^{-1}$.} \proof Consider the symplectic cylinder $\Cyl$ obtained as a product of the disk $\{ax_1 + b y_1 \ \ |\ \ a^2 + b ^2 \leq \alpha_1^{-1}\}$ and the vector space $\langle x_2, y_2, x_3, y_3, ...\rangle$. Then $\Cyl \supset E \supset B_r$, where $B_r$ is a ball of radius $r:=\alpha_1^{-\nicefrac 1 2}$. {\bf \purple By Gromov's non-squeezing theorem, $\capa_G(\Cyl)=\capa_G(B_r)=\pi \alpha_1^{-1}$. By monotonicity of $\capa_G$, its capacity is the same.} \endproof \newpage {\bf \blue Symplectic cylinders and capacity} \definition Let $(V, \omega)$ be a symplectic space, and $W\subset V$ a 2-dimensional symplectic space. Denote the symplectic orthogonal by $W^{\bot_\omega}$. {\bf \blue Cylinder} $\Cyl_E\subset (V, \omega)$ is a product $\Cyl_E:= E \times W^{\bot_\omega}$ of an ellipsoid $E\subset W$ and $W^{\bot_\omega}$. {\bf \purple We consider $\Cyl_E$ as a symplectic cylinder in $V$.} The following claim follows from the result about the Gromov capacity of ellipsoids. \corollary Let $\phi:\; V \arrow W$ be an invertible linear map preserving the Gromov capacities of all ellipsoids. {\bf \red Then $\phi$ preserves the symplectic capacities of cylinders.} \proof The cylinder $\Cyl_E:=\{ax_1 + b y_1 \ \ |\ \ a^2 + b ^2 \leq r\}$ can be obtained as a union of an increasing family of ellipsoids \[ E_\alpha=\{v= \sum a_i x_i +b_i y_i \ \ | \ \ a_1^2 + b_1^2 + \sum_{i=2}^\infty \alpha(a_i^2+b_i^2)< r\}, \alpha \arrow \infty \] of the same capacity. For any relatively compact symplectic ball $B_r\subset \Cyl_E$, it is contained in one of these ellipsoids, hence $\pi r^2$ is smaller than the capacity of the ellipsoid. Then $\capa_G (\Cyl_E) \leq \capa_G(E_\alpha) = \capa_G(E)$. \endproof \newpage {\bf \blue Fake cylinders} \definition Let $W_1\subset V$ be a coisotropic subspace of codimension 2, and $W\subset V$ a complementary subspace, $W_1 \oplus W=V$. Consider an ellipsoid $E \subset W$. The product $W_1 \times E\subset V$ is called {\bf \blue a fake cylinder}. \theorem {\bf \red A fake cylinder $Z:=W_1 \times E \subset V$ is symplectomorphic to $\R^{2n}$ with the usual symplectic structure}. \pstep Choose a symplectic basis in $V$ in such a way that $W_1=\langle x_1, x_2, x_3, y_3, x_4, y_4, ...\rangle$. Projecting $E$ to $\langle y_1, y_2\rangle$ along $W_1$, we obtain an ellipsoid $E' \subset \langle y_1, y_2\rangle$. Clearly, $W_1 \times E= W_1 \times E'$, hence {\bf \purple we can assume that $E$ is an ellipsoid in the space $\langle y_1, y_2\rangle$.} {\bf \green Step 2:} To finish the proof if would suffice to show that $Z= \langle x_1, x_2\rangle \times E$, where $E \subset \langle y_1, y_2\rangle$, is symplectomorphic to $\R^4$ with the usual symplectic structure. Consider the projection $Z\arrow E$ along $\langle x_1, x_2\rangle$. The form $\omega$ identifies the fibers of this projection with $T^*E= \langle y_1, y_2\rangle$. Therefore, {\bf \purple $Z$ is symplectomorphic to $T^*E$ with the usual symplectic structure.} Since $E$ is diffeomorphic to $\R^2$, the manifold $Z$ is symplectomorphic to $T^* \R^2$. \endproof \newpage {\bf \blue Proof of the linear version of Ekeland-Hofer} \corollary Let $V, \omega$ be a symplectic vector space, and $\phi:\; V \arrow V$ a linear map which preserves the symplectic capacity of all ellipsoids. {\bf \red Then $\phi$ maps coisotropic spaces of codimension 2 to coisotropic spaces}. \proof Let $Z\subset V$ be a cylinder or a fake cylinder, $Z= W \times E$. The space $W$ can be reconstructed from $E$ as follows: it is a set of all $v\in V$ such that $\lambda v \in Z$ for all $\lambda\in \R$. It is coisotropic if and only if $Z$ is a fake cylinder. {\bf \purple Since $\phi$ preserves the capacity, and capacity is finite for cylinders and infinite for fake cylinders, $\phi$ maps the cylinders to cylinders, and fake cylinders to fake cylinders}. \endproof The linear version of Ekeland-Hofer is implied by the following lemma. \lemma Let $V, \omega$ be a symplectic space, and $\phi:\; V \arrow V$ a linear map which takes coisotropic subspaces of codimension 2 to coisotropic subspaces. {\bf \red Then $\phi^*\omega= \const \cdot \omega$, that is, $\phi$ is a composition of homothety and a symplectomorphsm.} \newpage {\bf \blue Linear maps preserving coisotropic spaces} \lemma Let $V, \omega$ be a symplectic space, and $\phi:\; V \arrow V$ a linear map which takes coisotropic subspaces of codimension 2 to coisotropic subspaces. {\bf \red Then $\phi^*\omega= \const \cdot \omega$, that is, $\phi$ is a composition of homothety and a symplectomorphsm.} \proof Consider the correspondence mapping a coisotropic subspace $W\subset V$ of codimension 2 to $W^{\bot_\omega}$. It is easy to see that $W$ is coisotropic if and only if $\Ann(W)$ is isotropic. {\bf \purple We reduced the lemma to the following.} \lemma Let $V$ be a vector space, and $\omega_1, \omega_2$ symplectic forms such that any 2-dimensional plane which is isotropic in $\omega_1$ is isotropic in $\omega_2$. {\bf \red Then $\omega_1$, $\omega_2$ are proportional}. \proof For all $x, y\in V$, we have $\omega_1(x, y)=0$ $\Leftrightarrow$ $\omega_2(x, y)=0$. Multiplying $\omega_2$ by a constant, we may assume that $\omega_1(a,b)=\omega_2(a,b)\neq 0$ for some $a, b \in V$. Then $\omega_1(a,c)=\omega_2(a,c)$ for any $c$; indeed, otherwise $\omega_1(a, c-tb)=0$ for appropriate $t$, and $\omega_2(a,c)\neq 0$. Applying the same argument the second time, we obtain that $\omega_1(c,c')=\omega_2(c,c')$ for any $c, c'\in V$. \endproof \end{document}