What is the geometric meaning of ECH capacities?

Last week, Keon Choi, Dan Cristofaro-Gardiner, David Frenkel, Vinicius Ramos, and I posted a new paper, Symplectic embeddings into four-dimensional concave toric domains. To explain what concave toric domains are, recall that if \Omega is a domain in the first quadrant of the plane, we define a (four-dimensional) “toric domain”

X_\Omega = \{z\in{\mathbb C}^2 \mid \pi(|z_1|^2,|z_2|^2)\in\Omega\}.

For example, if \Omega is a right triangle with vertices (0,0), (a,0), (0,b), then X_\Omega is an ellipsoid, which we denote by E(a,b). We also define the ball B(a)=E(a,a). If \Omega is a rectangle with vertices (0,0), (a,0), (0,b), (a,b), then X_\Omega is the polydisk which we denote by P(a,b). Define a “concave toric domain” to be a domain X_\Omega where \Omega is bounded by the axes and the graph of a convex function [0,a]\to [0,b] for some positive real numbers a,b. So an ellipsoid is a concave toric domain, but a polydisk is not. The (nondisjoint) union of two ellipsoids is also an example of a concave toric domain.

In the paper we computed the ECH capacities of all concave toric domains, and obtained some applications to symplectic embedding problems. For now I will skip the discussion of the symplectic embedding problems, and focus on the following question: What is the geometric meaning of ECH capacities?

For a concave toric domain X_\Omega, we showed that there is a nonincreasing sequence of nonnegative real numbers a_1,a_2,\ldots such that the k^{th} ECH capacity of X_\Omega is given by

c_k(X_\Omega) = c_k\left(\coprod_iB(a_i)\right).

It turns out after the fact that the numbers a_i can be described geometrically as follows. First, a_1 is the largest real number a such that there exists a symplectic embedding B(a)\to X_\Omega. Then a_2 is, roughly speaking, the largest real number a such that there exists a symplectic embedding of int (B(a)) into the remaining space. And so forth. That is the numbers a_1,a_2,\ldots measure the size of the balls in a “greedy ball packing” of X_\Omega.

To make this precise, if X is any domain, we can define a_1(X) to be the supremum over a such that there exists a symplectic embedding B(a)\to X.  We can then inductively define a_{k+1}(X) to be the limit, as a_1',\ldots,a_k' approach a_1(X),\ldots,a_k(X) from below, of the maximum of a_1(X\setminus \phi(\coprod_{i=1}^k B(a_i'))), where the maximum is over symplectic embeddings \phi:\coprod_{i=1}^kB(a_i')\to X.

We now know that if X is a concave toric domain, then

c_k(X) = c_k\left(\coprod_{i=1}^\infty B(a_i(X))\right).

For which other domains X is this true? Note that the left hand side of the above equation is always greater than or equal to the right hand side, by the monotonicity of ECH capacities. However the reverse inequality need not be true.

The first counterexample I could find is the polydisk P(2,3). Here, if I am not mistaken, the sequence of numbers a_i is (2,2,1,1,1,1,0,\ldots). For k=4, we have c_4(P(2,3))=7, but c_4(\coprod_iB(a_i))=6.

So I guess we have more work to do to understand what ECH capacities are measuring in general.

 

Advertisements
This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s