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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 $\rho$.

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    $\begingroup$ What is the capacity you're referring to? Could you give a reference? Could you also provide some motivation for the question? In particular, why should the exponent be exactly $n$? Finally, have you tried computing any reasonable example (e.g. tori, products...)? $\endgroup$
    – Marco Golla
    Commented Jan 4, 2014 at 0:51
  • $\begingroup$ In the revised version I wrote the definition of symplectic capacity. Moreover the true exponent is not $n$ but is $2n$, the dimension of total space. I edited it know $\endgroup$
    – Ali Taghavi
    Commented Jan 4, 2014 at 7:28
  • $\begingroup$ Thank you for your comment on the "exponent".motivating by $M=S^{1}$,(and $M=\mathbb{T}^{n}$) we observe that the true power must be "2", otherwise the above limit goes to 0 or infinity. $\endgroup$
    – Ali Taghavi
    Commented Jan 4, 2014 at 8:21
  • $\begingroup$ I apologize for several changing the true exponent. it was because of my miss computation. the true exponent is 2n , because the volume of a $\rho$-disk in $\mathbb{R}^{2n}$ is of order $\rho^{2n}$. On the other hand a symplectic embedding preserve the volume $\endgroup$
    – Ali Taghavi
    Commented Jan 4, 2014 at 10:13

1 Answer 1

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Both limits (1) and (2) are equal to $C(1)/V(1)$ because of the homogeneity of the volume and the symplectic capacity. Namely, the symplectic form is homogeneous of degree $1$ with respect to dilations on the tangent bundle and so it makes no difference what the radius of your disc bundle is.

What is then geometric meaning of $C(1)/V(1)$ ? That's an interesting and difficult question which falls under the topic of capacity-volume inequalities (something that is just taking off). I'd suggest looking at flat tori first. The reason is that if you consider the universal cover, the lattice, and the ellipsoid that defines your unit disc, then if you have a lattice basis inside the ellipsoid, you also have a symplectic unit ball inside the unit tangent bundle of your torus : just look at the moment map of the torus action on the unit ball: it's image is (or can be made to be) the simplex formed by the origin and your lattice basis. You will probably be able to give a number-geometric interpretation to $C(1)/V(1)$ in this case.

By the way, the original capacity-volume problem is

The Viterbo Conjecture. The symplectic capacity (any capacity!) of a convex body $K \subset \mathbb{R}^{2n}$ whose volume is the same as that of the $2n$-dimensional Euclidean unit ball is less than or equal to $\pi$.

Note that this is obviously true for the Gromov width, but it is open for every other capacity. There is a chance this conjecture is true for the Hofer-Zehnder capacity, which can be more easily described as the least of the actions of all closed characteristics on the boundary of $K$. There are some partial results on this conjecture in the first version of this paper, but beware, by a recent result of Artstein-Avidan, Ostrover, and Karasev, a positive answer to Viterbo's conjecture would also prove the Mahler conjecture (corollary: this can't be easy!).

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  • $\begingroup$ thank you very much for your interesting answer. I apologize if my question is elementary; is it obvious that the capacity is homogeneous of order 1? assuming this, what is the the true exponent in definition of $C(r)$, $n$ or $2n$, because $V(r)$ is probably of order $n$(and what is reason for this last statement?) $\endgroup$
    – Ali Taghavi
    Commented Jan 8, 2014 at 16:38
  • $\begingroup$ $V(r)$ has order $n$: when you dilate in the tangent bundle you only dilate in the direction of the velocities (half the dimension of the total space). Every capacity satisfies $c(M, \lambda \omega) = \lambda c(M, \omega)$ for $\lambda > 0$ (and, yes, this is basically immediate for Gromov width). Since dilation in the velocities dilates the symplectic form, you need to use the exponent $n$ on both the capacity and the volume so that the question be meaningful. $\endgroup$
    – alvarezpaiva
    Commented Jan 8, 2014 at 16:51
  • $\begingroup$ thank you very much for your comment, could you please introduce me a reference with minimum necessary background? $\endgroup$
    – Ali Taghavi
    Commented Jan 8, 2014 at 17:07
  • $\begingroup$ The "classic" text is McDuff, Salamon "Introduction to symplectic topology". $\endgroup$
    – alvarezpaiva
    Commented Jan 8, 2014 at 17:09
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    $\begingroup$ I don't want to sound mean spirited, but posting questions which should be research level and knowing only the basic definitions may not lead to interesting exchanges. Pick up McDuff and Salamon, it's a great book. $\endgroup$
    – alvarezpaiva
    Commented Jan 8, 2014 at 17:34

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