## Some open problems on symplectic embeddings and the Weinstein conjecture

Today was the first meeting this semester of the student Floer theory seminar.  I introduced some open problems and conjectures, at least some of which should be approachable.

The problems I discussed fall under two general themes:

1. SYMPLECTIC EMBEDDINGS: Given two symplectic manifolds $(X_0,\omega_0)$ and $(X_1,\omega_1)$ of dimension $2n$, typically with boundary, when does there exist a symplectic embedding of one into the other?

2. QUANTITATIVE REFINEMENTS OF THE WEINSTEIN CONJECTURE: Given a closed contact manifold $(Y^{2n-1},\lambda)$, does there exist a Reeb orbit with an upper bound on the length depending only on $vol(Y,\lambda)$ and the contact structure $(Y,\xi)$?  (This previous post has some discussion of this question when $n=2$.)

When $n=2$, there are new tools with which to study these problems, namely the “ECH spectrum” and “ECH capacities”.  The ECH spectrum of a contact three-manifold $(Y,\lambda)$ (with nonvanishing ECH contact invariant) is a sequence of numbers

$0 = c_0(Y,\lambda) < c_1(Y,\lambda) \le c_2(Y,\lambda) \le \cdots \le \infty$

This is relevant to quantitative refinements of the Weinstein conjecture, because by definition there exists a Reeb orbit of length at most $c_1(Y,\lambda)$.

The ECH capacities of a symplectic four-manifold $(X,\omega)$, typically with boundary, are a sequence of numbers

$0 = c_0(X,\omega) < c_1(X,\omega) \le c_2(X,\omega) \le \cdots \le \infty$

These are relevant to symplectic embeddings, because if $(X_0,\omega_0)$ symplectically embeds into $(X_1,\omega_1)$, then $c_k(X_0,\omega_0) \le c_k(X_1,\omega_1)$ for all $k$.

The relation between ECH capacities and ECH spectrum is that if $(X,\omega)$ is a symplectic four-manifold with boundary such that $\omega$ is exact, and there exists a contact form $\lambda$ on $Y=\partial X$ such that $d\lambda = \omega|_Y$, then $c_k(X,\omega) = c_k(Y,\lambda)$.

To give some examples:

The ECH capacities of the ellipsoid

$E(a,b) = \{(z_1,z_2)\in{\mathbb C}^2 \mid \frac{\pi |z_1|^2}{a} + \frac{\pi |z_2|^2}{b} \le 1\}$

are described as follows: $c_k(E(a,b))$ is the minimum of $am+bn$, where $m,n$ are natural numbers such that the triangle in the plane bounded by the $x$ and $y$ axes and the line through $(m,n)$ with slope $-b/a$ contains at least $k+1$ lattice points.

The ECH capacities of the polydisk

$P(a,b) = \{(z_1,z_2)\in{\mathbb C}^2 \mid \pi |z_1|^2 \le a, \pi |z_2|^2 \le b \}$

are described as follows: $c_k(P(a,b))$ is the minimum of $am+bn$, where the rectangle bounded by the axes together with horizontal and vertical lines through $(m,n)$ contains at least $k+1$ lattice points.

Open problem. Generalize to compute the ECH capacities of

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

where $\Omega$ is a reasonable domain in the first quadrant of the plane.  I think I know how to approach this, but it would be nontrivial to work out the details.

We can also compute the ECH capacities of a disjoint union:

$c_k(\coprod_{i=1}^n(X_i,\omega)) = \max\{\sum_{i=1}^n c_{k_i}(X_i,\omega_i) \mid \sum_{i=1}^n k_i = k\}.$

Now how good are the symplectic embedding obstructions given by ECH capacities?

1. Dusa McDuff showed the ECH capacities give a sharp obstruction to symplectically embedding a disjoint union of ellipsoids into an ellipsoids, i.e. the interior of the former symplectically embeds into the latter if and only if $c_k$ of the former is less than or equal to $c_k$ of the latter for all $k$.

2. It follows from McDuff’s arguments and work of Dorothee Mueller that ECH capacities likewise give a sharp obstruction to symplectically embedding an ellipsoid into a polydisk.

3. ECH capacities do not give a sharp obstruction to embedding a polydisk into an ellipsoid;  the much older Ekeland-Hofer capacities give stronger obstructions in some cases.

4. I do not know whether ECH capacities give a sharp obstruction to symplectically embedding a polydisk into a polydisk.

Conjecture. The Ekeland-Hofer capacities of a star-shaped domain in ${\mathbb R}^4$ agree with the linearized contact homology analogue of the “full ECH capacities”. I checked this for ellipsoids and polydisks.

For more examples, let $\|\cdot\|$ be a norm on ${\mathbb R}^2$ and let $\|\cdot\|^*$ denote its dual norm on $({\mathbb R}^2)^*$. Define

$X_{\|\cdot\|^*} = \{(x,\xi) \in T^*T^2 \mid \|\xi\|^*\le 1\}.$

Then it is known that $c_k(X_{\|\cdot\|^*})$ is the minimum of $\ell_{\|\cdot\|}(\Lambda)$, where $\Lambda$ ranges over convex polygons in the plane that enclose at least $k+1$ lattice points, and $\ell_{\|\cdot\|}$ denotes length with respect to the norm $\|\cdot\|$.  We can now ask when one of these manifolds symplectically embeds into another.

Optimistic conjecture: $X_{\|\cdot\|_0^*}$ symplectically embeds into $X_{\|\cdot\|_1^*}$ if and only if there exists $A \in SL_2{\mathbb Z}$ such that $A^*\|\cdot\|_0\le \|\cdot\|_1$.  (“If” is an easy exercise, “only if” is the optimistic part.)

By the theory of ECH capacities, this would follow from:

Stronger conjecture: $c_k(X_{\|\cdot\|_0^*}) \le c_k(X_{\|\cdot\|_1^*})$ for all $k$ if and only if there exists $A \in SL_2{\mathbb Z}$ such that $A^*\|\cdot\|_0\le \|\cdot\|_1$.  (Again, “if” is easy.)

Now this stronger conjecture is purely combinatorial, so you can sit down, maybe with a computer, and try checking it.

Project. The definition of the ECH capacities (which I did not explain here) can be easily modified in various ways using other versions of contact homology, and in particular for higher dimensional symplectic manfolds.  Explore these capacities and see if they give decent obstructions, e.g. by testing them on six-dimensional ellipsoids and comparing with work of Felix Schlenk.

Going back toward the quantitative Weinstein conjecture, we have:

Conjecture. If $(Y,\lambda)$ is a closed contact 3-manifold all of whose ECH spectrum is finite, then

$\lim_{k\to\infty}\frac{c_k(Y,\lambda)^2}{k} = 2 vol(Y,\lambda).$

There is overwhelming evidence for this conjecture, and I expect that it can be proved if one carefully studies the estimates in Taubes’s proof of the Weinstein conjecture.

By playing with the examples above, one might be tempted to guess that in fact,

$\frac{c_k(Y,\lambda)^2}{k} \le 2 vol(Y,\lambda)$

for all $k$. In particular, this would imply that there is always a Reeb orbit of length at most $\sqrt{2 vol(Y,\lambda)}$.  Unfortunately, the above guess is is disproved by the following example: let $Y$ be a circle bundle over $S^2$ of Euler class $e>0$ with a prequantization contact form. Then $vol(Y)=4\pi^2 e$, and $c_1(Y)=2\pi e$, which is bigger than $\sqrt{2 vol(Y)}$ when ${e} > {2}$. Note however that this example still has a Reeb orbit of length $2\pi$, which is less than $\sqrt{2 vol(Y)}$.  (Also, there is a variant of the ECH spectrum which detects this.)  So I still don’t know any counterexample to:

Conjecture. A closed contact three-manifold $(Y,\lambda)$ always has a Reeb orbit of length at most $\sqrt{2 vol(Y,\lambda)}$.

Project. See whether or not this conjecture is true for contact forms supported by an open book.

I should point out that we do know the ECH capacities of $X_\Omega$ when $\Omega$ is a convex domain that does not intersect the axes. In this case $X_\Omega$ is symplectomorphic to $X_{\|\cdot\|^*}$ where the unit ball of $\|\cdot\|^*$ is a translate of $\Omega$, so its ECH capacities are given by the theorem stated above. If $\Omega$ intersects the axes in one or two points then one will get the same answer by a limiting argument. I suspect that one will get the same answer even if $\Omega$ includes a segment of an axis. It is easy to check that this is true if $\Omega$ is a rectangle or a right triangle whose legs are on the axes (corresponding to an polydisk or ellipsoid), but proving this in general, if it is true, would require more work. I plan to say more about ECH capacities of $X_\Omega$ in a future post.