Comparison with Ekeland-Hofer

Helmut Hofer asked me how the symplectic embedding obstructions in “Beyond ECH capacities” compare to the obstructions given by Ekeland-Hofer capacities, for symplectic embeddings between ellipsoids and polydisks. I happened to know the answer to this, using something which I chose to leave out of the paper (too many things going on, needed to try to stay focused!), so I would like to explain it here. First, the answer is the following:

  • For symplectic embeddings of (four-dimensional) ellipsoids into ellipsoids or polydisks, ECH capacities give sharp obstructions (shown by McDuff), while Ekeland-Hofer capacities are often weaker.
  • For a polydisk P(a,b) into another polydisk P(a',b') where a\ge b and a'\ge b', Ekeland-Hofer only tells us that b\le b'. ECH capacities say a bit more but are not very good. “Beyond ECH capacities” gives sharp obstructions in some cases.
  • For a polydisk P(a,b) into an ellipsoid E(c,d), ECH capacities are not very good, and sometimes weaker than Ekeland-Hofer capacities. Ekeland-Hofer gives a sharp obstruction when a=b (and a less good obstruction when a\neq b). “Beyond ECH capacities” can recover this (the sharp obstruction when a=b; I haven’t checked that I can recover all information given by all Ekeland-Hofer capacities when a\neq b).

I would now like to explain this last point. First let’s change the notation: given a,b\ge 1, we would like to find the infimum of c such that the polydisk P(a,1) symplectically embeds into the ellipsoid E(bc,c). Observe that P(a,1) trivially embeds into E(bc,c) by inclusion if c\ge 1+a/b. When a=1, the converse is true:

Theorem 1. If b\ge 1 and P(1,1) symplectically embeds into E(bc,c), then c\ge 1+1/b.

Here’s how to prove this using Ekeland-Hofer capacities. Let’s denote the k^{th} Ekeland-Hofer capacity by c_k. (Usually I use this symbol to denote ECH capacities, but we won’t be talking about ECH capacities in this post.) The Ekeland-Hofer capacities of P(a,1) for a\ge 1 are given by

c_k(P(a,1)) = k.

On the other hand, c_k(E(bc,c)) is the k^{th} entry in the list of all positive integer multiples of bc or c, written in increasing order with repetitions. It follows that

c_{k + \lfloor k/b \rfloor}(E(bc,c)) = kc.

So if P(1,1) symplectically embeds into E(bc,c), then by the monotonicity property of Ekeland-Hofer capacities, for every positive integer k we have

k + \lfloor k/b \rfloor \le kc.

Taking k large gives 1 + 1/b \le c as desired. Now how do we recover this from “Beyond ECH capacities”? Doing something rather crude with the methods in that paper, which I will explain below, one obtains the following:

Theorem 2. Suppose P(a,1) symplectically embeds into P(bc,c) where a,b\ge 1. Suppose also that

\sqrt{a/2} + \sqrt{1/(2a)} \le \sqrt{b} + 1/\sqrt{b}.

Then

2c \ge 1 + b^{-1} + \sqrt{1 + b^{-2}} + a(1 + b^{-1} - \sqrt{1 + b^{-2}}).

For example, if a=1, then we obtain c \ge 1 + b^{-1}, recovering Theorem 1.  Another example is that if b=1, then we obtain

c \ge (1 + 1/\sqrt{2}) + (1-1/\sqrt{2})a

for a\le 3+2\sqrt{2}. This is nontrivial, but weaker than Theorem 1.2 in “Beyond ECH capacities” when a>1.

Now I will explain the proof of Theorem 2, assuming as a prerequisite the statement of Theorem 1.18 in “Beyond ECH capacities”. We will need the following immediate corollary of the latter theorem:

Theorem 3. Let X_\Omega and X_{\Omega'} be convex toric tomains, and suppose that X_\Omega symplectically embeds into X_{\Omega'}. Let \Lambda' be a convex generator which is minimal for X_{\Omega'}. Then there exists a convex generator \Lambda such that

I(\Lambda) = I(\Lambda'),

A_{\Omega}(\Lambda) \le A_{\Omega'}(\Lambda'),

and

x(\Lambda) + y(\Lambda) \ge x(\Lambda') + y(\Lambda').

(To address a question of Dan C-G: You can also say that I - x - y defines a filtration on the ECH chain complex which is preserved by the cobordism map. However I’m not sure if the cobordism map necessarily induces an isomorphism on the homology of the associated graded, because I don’t know if this filtration will be preserved by the relevant chain homotopies. There are also variants of this filtration to play with coming from J_0 and J_+. An interesting topic to think about later.)

Here is how to deduce Theorem 2 from Theorem 3. Let \Lambda' be a convex generator which is minimal for E(bc,c) and has very large ECH index. Let us rescale this so that the area under the curve is b/2. Then the rescaled \Lambda' is approximately a straight line from (0,1) to (b,0). Now the convex generator \Lambda provided by Theorem 2, after rescaling by the same factor, is a curve from (0,y) to (x,0), for some positive real numbers x and y, which is the graph of a nonincreasing concave function, and the area under this curve is approximately b/2. It follows that

xy/2 \le b/2 \le xy,

up to some error which can be made arbitrarily small by taking the ECH index of \Lambda to be sufficiently large. The other two inequalities in Theorem 2 then tell us that, also up to a small error, we have

x+ay \le bc

and

x+y \ge b+1.

It follows that

bc\ge \min\{x+ay | xy/2\le b/2 \le xy, x+y\ge b+1\}.

It is now an exercise in undergraduate multivariable calculus to compute the minimum on the right hand side. If

\sqrt{a/2} + \sqrt{1/(2a)} \ge \sqrt{b} + 1/\sqrt{b},

then the minimum is \sqrt{2ab}. Thus we conclude that c\ge \sqrt{2a/b}. This is just the volume constraint vol(P(a,1)) \ge vol(E(bc,c)). On the other hand, if

\sqrt{a/2} + 1/\sqrt{1/(2a)} \ge \sqrt{b} + 1/\sqrt{b},

then the minimum is (b+1+\sqrt{b^2+1})/2 + a(b+1-\sqrt{b^2+1})/2, which proves Theorem 3.

One last remark: the asymptotics of the symplectic embedding obstructions coming from ECH capacities for large ECH index just recover the volume constraint. The example above shows that the asymptotics of the obstruction in Theorem 2 for large ECH index (and also the Ekeland-Hofer capacities for large k) sometimes say more.      

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