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\begin{document}
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\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} 
