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Let $(M,g)$ be a compact Riemannian manfold with $dim M=n$. This gives us a unique sympletic

structure $(TM, \omega)$ and a unique volum form $\Omega=\omega^{n}$ on $TM$.

For every $r>0$, let $D_{r}(M)$ be the open Disc bundle on $M$, with radius $r$. Namely $D_{r}(M)$

  is the subset of $TM$ which contains all vectors with length $<r$.

Define:

${C(r)=(\text{sympletic capacity of $D_{r}(M)$}})^{2n}$

$V(r)$=The Volum of $D_{r}(M)$ with respect to $\Omega$

Question :

1)Does $\lim_{r\to \infty} C(r)/V(r)$ exist? And what is its geometric interpretation?

2)Does $\lim_{r\to 0} C(r)/V(r)$ exist? And what is its geometric interpretation?

Note that the symplectic capacity of a sympletic manifold $N$of dimension 2n defined as follows:

$\sup\; \{\rho \mid \text{there is a symplectic embedding from $B_{\rho}(0)\subset \mathbb{R}^{2n}$ to $N$}\}$.

By $B_{\rho}(0)$ I mean the disc around the origin with radius $\rho$.

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold. Using the metric to identify the tangent and cotangent bundles defines a natural symplectic structure on the tangent bundle, $(TM, \omega)$, and a natural volume form $\Omega=\omega^{n}$ on $TM$.

For every $r>0$, let $D_{r}(M)$ be the open disc bundle on $M$, with radius $r$. Namely $D_{r}(M)$ is the subset of $TM$ formed by all tangent vectors with length smaller than $r$.

Define:

${C(r)=(\text{Gromov width of $D_{r}(M)$}})^{2n}$

$V(r)$=The Volum of $D_{r}(M)$ with respect to $\Omega$

Question :

1. Does $\lim_{r\to \infty} C(r)/V(r)$ exist? And what is its geometric interpretation?

2. Does $\lim_{r\to 0} C(r)/V(r)$ exist? And what is its geometric interpretation?

Recall that the Gromov width of a symplectic manifold $N$ of dimension 2n defined as follows:

$\sup\; \{\rho \mid \text{there is a symplectic embedding from $B_{\rho}(0)\subset \mathbb{R}^{2n}$ to $N$}\}$.

By $B_{\rho}(0)$ I mean the disc around the origin with radius $\rho$.

Let $(M,g)$ be a compact Riemannian manfold with $dim M=n$. This gives us a unique sympletic

structure $(TM, \omega)$ and a unique volum form $\Omega=\omega^{n}$ on $TM$.

For every $r>0$, let $D_{r}(M)$ be the open Disc bundle on $M$, with radius $r$. Namely $D_{r}(M)$

is the subset of $TM$ which contains all vectors with length $<r$.

Define:

${C(r)=(\text{sympletic capacity of $D_{r}(M)$}})^{2}$

$V(r)$=The Volum of $D_{r}(M)$ with respect to $\Omega$

Question:

Does $\lim_{r\to \infty} C(r)/V(r)$ exist? And what is its geometric interpretation?

Note that the symplectic capacity of a sympletic manifold $N$of dimension 2n defined as follows:

$\sup\; \{r \mid \text{there is a symplectic embedding from $B_{r}(0)\subset \mathbb{R}^{2n}$ to $N$}\}$.

By $B_{r}(0)$ I mean the disc around the origin with radius $r$.

Let $(M,g)$ be a compact Riemannian manfold with $dim M=n$. This gives us a unique sympletic

structure $(TM, \omega)$ and a unique volum form $\Omega=\omega^{n}$ on $TM$.

For every $r>0$, let $D_{r}(M)$ be the open Disc bundle on $M$, with radius $r$. Namely $D_{r}(M)$

is the subset of $TM$ which contains all vectors with length $<r$.

Define:

${C(r)=(\text{sympletic capacity of $D_{r}(M)$}})^{2n}$

$V(r)$=The Volum of $D_{r}(M)$ with respect to $\Omega$

Question  :

1)Does $\lim_{r\to \infty} C(r)/V(r)$ exist? And what is its geometric interpretation?

2)Does $\lim_{r\to 0} C(r)/V(r)$ exist? And what is its geometric interpretation?

Note that the symplectic capacity of a sympletic manifold $N$of dimension 2n defined as follows:

$\sup\; \{\rho \mid \text{there is a symplectic embedding from $B_{\rho}(0)\subset \mathbb{R}^{2n}$ to $N$}\}$.

By $B_{\rho}(0)$ I mean the disc around the origin with radius $\rho$.

Let $(M,g)$ be a compact Riemannian manfold with $dim M=n$. This gives us a unique sympletic

structure $(TM, \omega)$ and a unique volum form $\Omega=\omega^{n}$ on $TM$.

For every $r>0$, let $D_{r}(M)$ be the open Disc bundle on $M$, with radius $r$. Namely $D_{r}(M)$

is the subset of $TM$ which contains all vectors with length $<r$.

Define:

${C(r)=(\text{sympletic capacity of $D_{r}(M)$}})^{2n}$

$V(r)$=The Volum of $D_{r}(M)$ with respect to $\Omega$

Question:

Does $\lim_{r\to \infty} C(r)/V(r)$ exist? And what is its geometric interpretation?

Note that the symplectic capacity of a sympletic manifold $N$of dimension 2n defined as follows:

$\sup\; \{r \mid \text{there is a symplectic embedding from $B_{r}(0)\subset \mathbb{R}^{2n}$ to $N$}\}$.

By $B_{r}(0)$ I mean the disc around the origin with radius $r$.

Let $(M,g)$ be a compact Riemannian manfold with $dim M=n$. This gives us a unique sympletic

structure $(TM, \omega)$ and a unique volum form $\Omega=\omega^{n}$ on $TM$.

For every $r>0$, let $D_{r}(M)$ be the open Disc bundle on $M$, with radius $r$. Namely $D_{r}(M)$

is the subset of $TM$ which contains all vectors with length $<r$.

Define:

${C(r)=(\text{sympletic capacity of $D_{r}(M)$}})^{2}$

$V(r)$=The Volum of $D_{r}(M)$ with respect to $\Omega$

Question:

Does $\lim_{r\to \infty} C(r)/V(r)$ exist? And what is its geometric interpretation?

Note that the symplectic capacity of a sympletic manifold $N$of dimension 2n defined as follows:

$\sup\; \{r \mid \text{there is a symplectic embedding from $B_{r}(0)\subset \mathbb{R}^{2n}$ to $N$}\}$.

By $B_{r}(0)$ I mean the disc around the origin with radius $r$.