## 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.