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# Recent progress on symplectic embedding problems in four dimensions

Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved February 15, 2011 (received for review December 11, 2010)

## Abstract

We survey some recent progress on understanding when one four-dimensional symplectic manifold can be symplectically embedded into another. In 2010, McDuff established a number-theoretic criterion for the existence of a symplectic embedding of one four-dimensional ellipsoid into another. This result is related to previously known criteria for when a disjoint union of balls can be symplectically embedded into a ball. Numerical invariants defined using embedded contact homology give general obstructions to symplectic embeddings in four dimensions which turn out to be sharp in the above cases.

Recall that a *symplectic manifold* is a pair (*X*,*ω*), where *X* is an oriented smooth manifold of dimension 2*n* for some integer *n*, and *ω* is a closed 2-form on *X* such that the top exterior power *ω*^{n} > 0 on all of *X*. The basic example of a symplectic manifold is with coordinates *z*_{j} = *x*_{j} + *iy*_{j} for *j* = 1,…,*n*, with the standard symplectic form [1]If (*X*_{0},*ω*_{0}) and (*X*_{1},*ω*_{1}) are two symplectic manifolds of dimension 2*n*, it is interesting to ask whether there exists a *symplectic embedding* *ϕ*: (*X*_{0},*ω*_{0}) → (*X*_{1},*ω*_{1})—i.e., a smooth embedding *ϕ*: *X*_{0} → *X*_{1} such that *ϕ*^{∗}*ω*_{1} = *ω*_{0}.

It turns out that the answer to this question is unknown, or only recently known, even for some very simple examples such as the following. If *a*_{1},…,*a*_{n} > 0, define the ellipsoid In particular, define the ball Also define the polydisk In these examples, the symplectic form is taken to be the restriction of *ω*_{std}.

An obvious necessary condition for the existence of a symplectic embedding *ϕ*: (*X*_{0},*ω*_{0}) → (*X*_{1},*ω*_{1}) is the volume constraint [2]where the volume of a symplectic manifold is defined by However, in dimensions greater than two, the volume constraint **2** is far from sufficient, even for convex subsets of , as shown by the famous:

There exists a symplectic embedding *B*(*r*) → *P*(*R*,∞,…,∞) if and only if *r* ≤ *R* (Gromov, ref. 1).

Let us now restrict to the case of dimension four. As we will see below, symplectic embedding problems are more tractable in four dimensions than in higher dimensions, among other reasons due to the availability of Seiberg–Witten theory. But symplectic embedding problems in four dimensions are still hard. For example, the question of when one four-dimensional ellipsoid can be symplectically embedded into another was answered only in 2010, by McDuff 2. (The analogous question in higher dimensions remains open.) To state the result, if *a* and *b* are positive real numbers, and if *k* is a positive integer, define (*a*,*b*)_{k} to be the *k*th smallest entry in the array , counted with repetitions. Denote the sequence ((*a*,*b*)_{k+1})_{k≥0} by . For example, If {*c*_{k}}_{k≥0} and are two sequences of real numbers indexed by nonnegative integers, the notation means that for every *k*≥0.

There exists a symplectic embedding int(*E*(*a*,*b*)) → *E*(*c*,*d*) if and only if (McDuff, ref. 2).

Note that given specific real numbers *a*, *b*, *c*, *d*, it can be a nontrivial number-theoretic problem to decide whether . For example, consider the special case where *c* = *d* (i.e., the problem of symplectically embedding an ellipsoid into a ball). By scaling, we can encode this problem into a single function *f* defined as follows: If *a* is a positive real number, define *f*(*a*) to be the infimum of the set of such that there exists a symplectic embedding int(*E*(*a*,1)) → *B*(*c*). Note that vol(*E*(*a*,*b*)) = *ab*/2, so the volume constraint implies that . By Theorem 1, McDuff and Schlenk (3) computed *f* explicitly (without using Theorem 1) and found in particular that

If , then

*f*is piecewise linear.The interval is partitioned into finitely many intervals, on each of which either

*f*is linear or .If

*a*≥(17/6)^{2}then .

The starting point for the proof of Theorem 1 is that, in ref. 4), the ellipsoid embedding problem is reduced to an instance of the *ball packing problem*: Given positive real numbers *a*_{1},…,*a*_{m} and *a*, when does there exist a symplectic embedding ? (Here and below, all of our balls are four-dimensional.) It turns out that the answer to this problem has been understood in various forms since the 1990s. The form that is the most relevant for our discussion is the following:

There exists a symplectic embedding if and only if [3]whenever (*d*_{1},…,*d*_{m},*d*) are nonnegative integers such that [4]

For example, consider the special case where all of the numbers *a*_{i} are equal, and say *a* = 1. Define *ν*(*m*) to be the supremum, over all symplectic embeddings of the disjoint union of *m* equal balls into *B*(1), of the fraction of the volume of *B*(1) that is filled.

The upper bounds on *ν*(*m*) for *m* = 2, 3, 5, 6, 7, 8 follow from Theorem 2 by taking (*d*_{1},…,*d*_{m},*d*) to be (1, 1, 1), (1, 1, 0, 1), (1, 1, 1, 1, 1, 2), (1, 1, 1, 1, 1, 0, 2), (2, 1, 1, 1, 1, 1, 1, 3), and (3, 2, 2, 2, 2, 2, 2, 2, 6), respectively. For the proof that the upper bounds in (a) are sharp, see ref. 5.

To prove (b), by Theorem 2, it is enough to show that, if (*d*_{1},…,*d*_{m},*d*) satisfy [**4**], then . To do so, consider two cases. First, if , then the Cauchy–Schwarz inequality gives . Second, if , then

Traynor (7) gives an explicit construction of a maximal symplectic packing of *B*(1) by *m* equal open balls when *m* is a perfect square or when *m* ≤ 6 (compare Proposition 7 below and see ref. 8 for an extensive discussion). Explicit constructions for *m* = 7, 8 are given by Wieck (9). However, explicit maximal packings are not known in general, and the proof of Theorem 2 is rather indirect.

## Symplectic Capacities from Embedded Contact Homology

We now explain how the “only if” parts of Theorems 1 and 2 can be recovered from some more general symplectic embedding obstructions.

A *symplectic capacity* is a function *c*, defined on some class of symplectic manifolds, with values in [0,∞], with the following properties:

(Monotonicity) If there exists a symplectic embedding (

*X*_{0},*ω*_{0}) → (*X*_{1},*ω*_{1}), then*c*(*X*_{0},*ω*_{0}) ≤*c*(*X*_{1},*ω*_{1}).(Conformality) If α is a nonzero real number, then

*c*(*X*,*αω*) = |*α*|*c*(*X*,*ω*).

See ref. 10 for a review of many symplectic capacities.

In ref. 11, embedded contact homology (ECH), which is reviewed later in this article, was used to define a sequence of symplectic capacities in four dimensions, called *ECH capacities*. If (*X*,*ω*) is a symplectic four-manifold (not necessarily closed or connected), its ECH capacities are a sequence of numbers [5]We denote the entire sequence by *c*_{•}(*X*,*ω*) = (*c*_{k}(*X*,*ω*))_{k≥0}. The following are some basic properties of the ECH capacities:

The ECH capacities satisfy the following axioms (11):

(Monotonicity)

*c*_{k}is monotone for each*k*.(Conformality)

*c*_{k}is conformal for each*k*.(Ellipsoid) .

(Disjoint Union)

In particular, the ECH capacities give sharp obstructions to symplectically embedding one ellipsoid into another, or a disjoint union of balls into a ball:

(a) If there is a symplectic embedding

*E*(*a*,*b*) →*E*(*c*,*d*), then .(b) If there is a symplectic embedding , then the inequalities

**3**hold.

(a) This implication follows immediately from the Monotonicity and Ellipsoid axioms in Theorem 4.

(b) First note that by the Ellipsoid axiom in Theorem 4, we have

*c*_{k}(*B*(*a*)) =*d*, where*d*is the unique nonnegative integer such that

Now suppose there is a symplectic embedding , and suppose (*d*_{1},…,*d*_{m},*d*) are nonnegative integers satisfying [**4**]. Let and and *k*^{′} = (*d*^{2} + 3*d*)/2. By hypothesis, *k* ≤ *k*^{′}. We then have where the first inequality holds by the Disjoint Union axiom, the second by Monotonicity, and the third by [**5**].

## Ball Packing

We now review the proof of Theorem 2 and related criteria for the existence of a symplectic embedding . By scaling, we may assume that *a* = 1.

The first step is to show that the existence of a ball packing is equivalent to the existence of a certain symplectic form on . There is a standard symplectic form *ω* on such that 〈*L*,*ω*〉 = 1, where *L* denotes the homology class of a line. With this symplectic form, , and there is a symplectic embedding . Now suppose there exists a symplectic embedding . We then have a symplectic embedding . We can now perform the “symplectic blowup” along (the image of) each of the balls *B*(*a*_{i}), which amounts to removing the interior of *B*(*a*_{i}), and then collapsing the fibers of the Hopf fibration on ∂*B*(*a*_{i}) to points, so that ∂*B*(*a*_{i}) is collapsed to the *i*th exceptional divisor. The result is a symplectic form *ω* on whose cohomology class is given by [6]where *E*_{i} denotes the homology class of the *i*th exceptional divisor, and PD denotes Poincaré duality. Also, the canonical class for this symplectic form (namely -*c*_{1} of the tangent bundle as defined using an *ω*-compatible almost complex structure) is given by [7]

To proceed, define to be the set of classes in that have square -1 and can be represented by a smoothly embedded sphere that is symplectic with respect to some symplectic form *ω* obtained from blowing up . Elements of are called “exceptional classes.” One can show that the set does not depend on the choice of *ω* as above. In fact, Li and Li (12) used Seiberg–Witten theory to show that consists of the set of classes *A* such that *A*^{2} = *A*·*K* = -1 and *A* is representable by a *smoothly* embedded sphere.

Let *a*_{1},…,*a*_{m} > 0, then the following are equivalent:

(a)⇒(b) follows from the blowup construction described above.

(b)⇒(a): It is shown in ref. 5 that, if [**6**] holds and if *ω* is homotopic through symplectic forms to a form obtained by blowing up along small balls, then one can “blow down” to obtain a ball packing. And it is shown in ref. 13 that any two symplectic forms on satisfying [**7**] are homotopic through symplectic forms.

(b)⇒(c) because *ω*^{2} > 0 and *ω* has positive pairing with every exceptional class.

(c)⇒(b) is proved in ref. 13. Actually, by Lemma 1 below, it is enough to prove the slightly weaker statement that if (c) holds then (*a*_{1},…,*a*_{m}) is in the closure of the set of tuples satisfying (b). This last statement follows from earlier work of McDuff (14, lemma 2.2) and Biran (15, theorem 3.2). The idea of the argument is as follows. Without loss of generality, *a*_{1},…,*a*_{m} are rational. Write . Let *ω*_{0} be a symplectic form on obtained by blowing up along small balls. We will see below that for some positive integer *n* there exists a connected, embedded, *ω*_{0}-symplectic surface *C* representing the class *nA*. Then, because *A*·*A* > 0 [by the first condition in (c)], the “inflation” procedure in ref. 14 (lemma 1.1) allows one to deform *ω*_{0} in a neighborhood of *C* to obtain a symplectic form *ω* with cohomology class [*ω*] = *ω*_{0} + *r*PD(*A*) for any *r* > 0. By taking *r* large and scaling, one can produce a symplectic form whose cohomology class is arbitrarily close to PD(*A*).

To find a surface *C* as above, choose a generic *ω*_{0}-compatible almost complex structure *J*. By the wall crossing formula for Seiberg–Witten invariants (16) and Taubes’s “SW⇒Gr” theorem (17), if is any class with *α*^{2} - *K*·*α*≥0 and *ω*_{0}·(*K* - *α*) < 0, then there exists a *J*-holomorphic curve *C* in the class α. These conditions are satisfied by *α* = *nA* when *n* is large. It turns out that the resulting holomorphic curve *C* is a connected embedded symplectic surface as desired, unless it includes an embedded sphere Σ of self-intersection -1 (or a multiple cover thereof) which does not intersect the rest of *C*. In this last case, Σ would represent an exceptional class with *A*·Σ < 0, contradicting the second condition in (c).

(d)⇒(c): Assume that (d) holds. The first part of (c) follows by an easy calculus exercise (see ref. 11) and is also a special case of a general relation between ECH capacities and symplectic volume discussed at the end of this article. To prove the rest of (c), let be an exceptional class. Because *A*^{2} = -1, we have . Furthermore, the adjunction formula implies that *K*·*A* = -1, so by [**7**] we have . Adding these two equations gives . We must have *d*≥0, because *A* is represented by an embedded surface which is symplectic with respect to *ω*_{0}. If all of the integers *d*_{i} are nonnegative as well, then (d) implies that as desired. If any of the integers *d*_{i} are negative, then replace them by 0 and apply (d) to obtain an even stronger inequality.

(a)⇒(d): This implication follows by invoking the ECH capacities as in Corollary 5. One can also prove this without using ECH capacities as follows. Suppose there is a ball packing as in (a), let *ω* be a symplectic form as in (b), and let *J* be a generic *ω*-compatible almost complex structure. Let (*d*_{1},…,*d*_{m},*d*) be nonnegative integers (not all zero) satisfying [**4**], and write . Then *α*^{2} - *K*·*α*≥0 by [**4**], and *ω*·(*K* - *α*) < 0 because *d*_{1},…,*d*_{m},*d* are nonnegative, so as in the proof of (c)⇒(b) there exists a *J*-holomorphic curve *C* representing the class α. Because this curve must have positive symplectic area with respect to *ω*, we obtain the inequality **3**.

As an alternative to the above paragraph, McDuff (2) proves that (c)⇒(d) by an algebraic argument using the explicit description of in ref. 12.

Theorem 2 now follows from Proposition 6, together with the following technical lemma:

There is a symplectic embedding if there is a symplectic embedding for every *λ* < 1.

It is shown in ref. 14 that any two symplectic embeddings are equivalent via a symplectomorphism of int(*B*(*a*)). Consequently, if there is a symplectic embedding for every *λ* < 1, then we can obtain a sequence of symplectic embeddings such that *ϕ*_{n} is the restriction of *ϕ*_{n+1}. The direct limit of the maps *ϕ*_{n} then gives the desired symplectic embedding .

## Ellipsoid Embeddings

We now explain McDuff’s proof of Theorem 1 using Theorem 2. By a continuity argument as in Lemma 1, we can assume without loss of generality that *a*/*b* and *c*/*d* are rational.

If *a* and *b* are positive real numbers with *a*/*b* rational, the *weight expansion* *W*(*a*,*b*) is a finite list of real numbers (possibly repeated) defined recursively as follows:

If

*a*<*b*, then*W*(*a*,*b*) = (*a*)∪*W*(*a*,*b*-*a*).*W*(*a*,*b*) =*W*(*b*,*a*).*W*(*a*,*a*) = (*a*).

For example, *W*(5,3) = (3,2,1,1). If *W*(*a*,*b*) = (*a*_{1},…,*a*_{m}), write . Ellipsoid embeddings are then related to ball packings as follows:

Suppose *a*/*b* and *c*/*d* are rational with *c* < *d*. (McDuff, ref. 4) Then there is a symplectic embedding int(*E*(*a*,*b*)) → *E*(*c*,*d*) if and only if there is a symplectic embedding [8]

We will only explain the easier direction—namely, why an ellipsoid embedding gives rise to a ball packing. For this purpose, consider the moment map defined by *μ*(*z*_{1},*z*_{2}) = *π*(|*z*_{1}|^{2},|*z*_{2}|^{2}). Call two subsets of “affine equivalent” if one can be obtained from the other by the action of and translations. Note that if *U*_{1}, *U*_{2} are affine equivalent open sets in the positive quadrant of , then *μ*^{-1}(*U*_{1}) and *μ*^{-1}(*U*_{2}) are symplectomorphic.

If *a*,*b* > 0, let Δ(*a*,*b*) denote the triangle in with vertices (0, 0), (*a*,0), and (0,*b*). Then *E*(*a*,*b*) = *μ*^{-1}(Δ(*a*,*b*)). If *a* < *b*, then Δ(*a*,*b*) is the union (along a line segment) of Δ(*a*,*a*) and a triangle which is affine equivalent to Δ(*a*,*b* - *a*). It follows by induction that if *W*(*a*,*b*) = (*a*_{1},…,*a*_{m}), then Δ(*a*,*b*) is partitioned into *m* triangles, such that the *i*th triangle is affine equivalent to Δ(*a*_{i},*a*_{i}). By Traynor (7), there is a symplectic embedding of int(*B*(*a*_{i})) into *μ*^{-1}(int(Δ(*a*_{i},*a*_{i}))). Hence there is a symplectic embedding [9]Likewise, int(Δ(*d*,*d*))∖Δ(*c*,*d*) is affine equivalent to int(Δ(*d* - *c*,*d*)), so there is a symplectic embedding and hence a symplectic embedding [10]If there is a symplectic embedding int(*E*(*a*,*b*)) → *E*(*c*,*d*), then composing it with the embeddings **9** and **10** gives a symplectic embedding as in [**8**].

The idea of the proof of Theorem 1 is to use the fact that the existence of an ellipsoid embedding is equivalent to the existence of a ball packing, and the fact that ECH capacities give a sharp obstruction to the existence of ball packings, to deduce that ECH capacities give a sharp obstruction to the existence of ellipsoid embeddings. To proceed with the details, if *a*_{•} and are sequences of real numbers indexed by nonnegative integers, define another such sequence by Note that the operation # is associative, and the Disjoint Union axiom of ECH capacities can be restated as

The “only if” part follows from Corollary 5(a). To prove the “if” part, assume without loss of generality that *a*/*b* and *c*/*d* are rational and *c* < *d*, and suppose that . By Proposition 7, we need to show that there exists a symplectic embedding as in [**8**]. By Theorem 2 and the calculation in Corollary 5(b), it is enough to show that To prove this inequality, first note that applying the Monotonicity axiom to the embedding **9** and using our hypothesis gives By the Disjoint Union axiom and the fact that the operation # respects inequality of sequences, it follows that On the other hand, applying Monotonicity to the embedding **10** gives By the above two inequalities, we are done.

The above is McDuff’s original proof of Theorem 1. Her subsequent proof in ref. 2 avoids using the monotonicity of ECH capacities (a heavy piece of machinery) as follows. The idea is to *define* the ECH capacities of any union of balls or ellipsoids by the Ellipsoid and Disjoint Union axioms, and then to algebraically justify all invocations of Monotonicity in the proof. For example, in the above argument for the “if” part of Theorem 1, in the first step, one needs to show that if *a*/*b* is rational then *c*_{•}(*B*(*a*,*b*)) ≤ *c*_{•}(*E*(*a*,*b*)). In fact, one can show algebraically that *c*_{•}(*B*(*a*,*b*)) = *c*_{•}(*E*(*a*,*b*)). To do so, by induction and the associativity of #, it is enough to show that if *a*/*b* is rational and *a* < *b* then . The proof of this fact may be found in ref. 2.

The proof of Theorem 1 generalizes to show that ECH capacities give a sharp obstruction to symplectically embedding any disjoint union of finitely many ellipsoids into an ellipsoid.

Theorem 1 does not directly generalize to higher dimensions. That is, if one defines to be the sequence of nonnegative integer linear combinations of *a*_{1},…,*a*_{n} in increasing order, then when *n* > 2 it is not true that int(*E*(*a*_{1},…,*a*_{n})) symplectically embeds into if and only if . In particular, Hind and Kerman (18) used methods of Guth (19) to show that *E*(1,*R*,*R*) symplectically embeds into *E*(*a*,*a*,*R*^{2}) whenever *a* > 3. However, if *R* is sufficiently large with respect to *a*, then .

## Embedded Contact Homology

In the ball packing story above, an important role was played by Taubes’s SW = Gr theorem, which relates Seiberg–Witten invariants of symplectic 4-manifolds to holomorphic curves. The ECH capacities are defined using an analogue of SW = Gr for contact 3-manifolds.

Let *Y* be a closed oriented 3-manifold. Recall that a *contact form* on *Y* is a 1-form *λ* on *Y* such that *λ*∧*dλ* > 0 everywhere. The contact form *λ* determines a *Reeb vector field* *R* characterized by *dλ*(*R*,·) = 0 and *λ*(*R*) = 1. A *Reeb orbit* is a closed orbit of *R*—i.e., a map for some *T* > 0, modulo reparametrization, such that *γ*^{′}(*t*) = *R*(*γ*(*t*)). The contact form *λ* is called “nondegenerate” if all Reeb orbits are cut out transversely in an appropriate sense. Generic contact forms *λ* are nondegenerate.

If *λ* is a nondegenerate contact form on *Y* as above, and if Γ∈*H*_{1}(*Y*), the *embedded contact homology* ECH_{∗}(*Y*,*λ*,Γ) is defined as follows. It is the homology of a chain complex ECC_{∗}(*Y*,*λ*,Γ) which is freely generated over (it can also be defined over but we will not need to do so). A generator is a finite set of pairs *α* = {(*α*_{i},*m*_{i})} consisting of distinct embedded Reeb orbits *α*_{i} together with positive integers *m*_{i}, such that *m*_{i} = 1 whenever *α*_{i} is hyperbolic (i.e., the linearized Reeb flow around *α*_{i} has real eigenvalues), and . The chain complex has a relative grading which is defined in ref. 20; the details of which are not important here.

To define the differential one chooses a generic almost complex structure *J* on with the following properties: *J* is -invariant, *J*(∂_{s}) = *R* where *s* denotes the coordinate, and *J* sends the contact structure Ker(*λ*)⊂*TY* to itself, rotating positively in the sense that *dλ*(*v*,*Jv*) > 0 for 0 ≠ *v*∈Ker(*λ*). If *α* = {(*α*_{i},*m*_{i})} and *β* = {(*β*_{j},*n*_{j})} are two chain complex generators, then the differential coefficient is a mod 2 count of *J*-holomorphic curves in which have “ECH index” equal to 1 and which converge as currents to as *s* → +∞ and to as *s* → -∞. Holomorphic curves with ECH index 1 have various special properties, one of which is that they are embedded (except that they may include multiply covered -invariant cylinders), hence the name “embedded contact homology.” For details see ref. 21 and the references therein. It is shown in ref. 22 that ∂^{2} = 0. Although the differential usually depends on the choice of *J*, the homology of the chain complex does not. In fact, it is shown by Taubes (23) that ECH_{∗}(*Y*,*λ*,Γ) is isomorphic to a version of Seiberg–Witten Floer cohomology of *Y* as defined in ref. 24.

The definition of ECH capacities only uses the case Γ = 0.

To define the ECH capacities, we need to recall four additional structures on embedded contact homology:

There is a chain map which counts

*J*-holomorphic curves of ECH index 2 passing through a generic point in (see ref. 25). The induced map on homology does not depend on*z*when*Y*is connected. If*Y*has*n*components, then there are*n*different versions of the*U*map.If

*α*= {(*α*_{i},*m*_{i})} is a generator of the ECH chain complex, define its*symplectic action*by It follows from the conditions on*J*that the differential decreases the symplectic action—i.e., if 〈∂*α*,*β*〉 ≠ 0 then . Hence for each , we can define ECH^{L}(*Y*,*λ*,Γ) to be the homology of the subcomplex spanned by generators with action less than*L*. It is shown in ref. 26 that this homology does not depend on*J*, although, unlike the usual ECH, it does depend strongly on*λ*.The

*empty set*of Reeb orbits is a legitimate generator of the ECH chain complex and, by the above discussion, it is a cycle. Thus we have a canonical elementLet (

*Y*_{+},*λ*_{+}) and (*Y*_{-},*λ*_{-}) be closed oriented 3-manifolds with nondegenerate contact forms. A*weakly exact symplectic cobordism*from (*Y*_{+},*λ*_{+}) to (*Y*_{-},*λ*_{-}) is a compact symplectic 4-manifold (*X*,*ω*) such that ∂*X*=*Y*_{+}-*Y*_{-}, the form*ω*on*X*is exact, and*ω*|_{Y±}=*dλ*_{±}. The key result which enables the definition of the ECH capacities is the following:Theorem 8.A weakly exact symplectic cobordism (

*X*,*ω*) as above induces maps for each with the following properties:(a) Φ

^{L}[∅] = [∅].(b) If

*U*_{+}is any of the*U*maps for*Y*_{+}, and if*U*_{-}is any of the*U*maps for*Y*_{-}corresponding to the same component of*X*, then Φ^{L}∘*U*_{+}=*U*_{-}∘Φ^{L}.

Idea of Proof:This theorem follows from a slight modification of the main result of ref. 26, as explained in ref. 11. The first step is to define a “completion” of

*X*by attaching cylindrical ends [0,∞) ×*Y*_{+}to the positive boundary and (-∞,0] ×*Y*_{-}to the negative boundary. One chooses an almost complex structure*J*on which is*ω*-compatible on*X*and which on the ends agrees with almost complex structures as needed to define the ECH of (*Y*_{±},*λ*_{±}).One would like to define a chain map ECC

_{∗}(*Y*_{+},*λ*_{+},0) → ECC_{∗}(*Y*_{-},*λ*_{-},0) by counting*J*-holomorphic curves in with ECH index 0. Considering ends of ECH index 1 moduli spaces would prove that this is a chain map. The conditions on*J*and the fact that we are restricting to Γ = 0 imply that this map would respect the symplectic action filtrations and satisfy property (a). To prove property (b), one would choose a path*ρ*in from a positive cylindrical end to a negative cylindrical end. Counting ECH index 1 curves that pass through*ρ*would then define a chain homotopy as needed to prove that Φ^{L}∘*U*_{+}=*U*_{-}∘Φ^{L}.Unfortunately, it is not currently known how to define Φ

^{L}by counting holomorphic curves as above because of technical difficulties caused by multiply covered holomorphic curves with negative ECH index (see ref. 20, section 5). However one can still define Φ^{L}and prove properties (a) and (b) by passing to Seiberg–Witten theory, using arguments from ref. 23.

## Definition of ECH Capacities

Let *Y* be a closed oriented 3-manifold with a contact form *λ*, and suppose that [∅] ≠ 0∈ECH_{∗}(*Y*,*λ*,0). We then define a sequence of real numbers as follows. Suppose first that *λ* is nondegenerate. If *Y* is connected, define If *Y* is disconnected, define *c*_{k}(*Y*,*λ*) the same way, but replace the condition *U*^{k}*η* = [∅] with the condition that every *k*-fold composition of *U* maps send *η* to [∅]. Finally, if *λ* is degenerate, one defines *c*_{k}(*Y*,*λ*) by approximating *λ* by nondegenerate contact forms.

Moving back up to four dimensions, define a (four-dimensional) *Liouville domain* to be a compact symplectic four-manifold (*X*,*ω*) such that *ω* is exact, and there exists a contact form *λ* on ∂*X* with *dλ* = *ω*|_{∂X}. In other words, (*X*,*ω*) is a weakly exact symplectic cobordism from a contact three-manifold to the empty set. For example, any star-shaped subset of is a Liouville domain. Here “star-shaped” means that the boundary is transverse to the radial vector field, and we take the standard symplectic form [**1**] as usual.

If (*X*,*ω*) is a Liouville domain, define its ECH capacities by where *λ* is any contact form on ∂*X* with *dλ* = *ω*|_{∂X}. Note here that *c*_{k}(∂*X*,*λ*) is defined because it follows from Theorem 8(a) that [∅] ≠ 0∈ECH_{∗}(∂*X*,*λ*,0). Also, *c*_{k}(*X*,*ω*) does not depend on *λ*, because changing *λ* will not change the ECH chain complex, and the fact that we restrict to Γ = 0 implies that changing *λ* does not affect the symplectic action filtration.

The Conformality axiom follows directly from the definition. The Ellipsoid and Disjoint Union axioms are proved by direct calculations in ref. 11. To prove the Monotonicity axiom, let (*X*_{0},*ω*_{0}) and (*X*_{1},*ω*_{1}) be Liouville domains and let *ϕ*: (*X*_{0},*ω*_{0}) → (*X*_{1},*ω*_{1}) be a symplectic embedding. Let *λ*_{i} be a contact form on ∂*X*_{i} with *dλ*_{i} = *ω*|_{∂Xi} for *i* = 0, 1. By a continuity argument we can assume without loss of generality that *ϕ*(*X*_{0})⊂ int(*X*_{1}) and that the contact forms *λ*_{i} are nondegenerate. Then (*X*_{1}∖*ϕ*(int(*X*_{1})),*ω*_{1}) defines a weakly exact symplectic cobordism from (∂*X*_{1},*λ*_{1}) to (∂*X*_{0},*λ*_{0}). It follows immediately from Theorem 8 that *c*_{k}(∂*X*_{1},*λ*_{1})≥*c*_{k}(∂*X*_{0},*λ*_{0}), because the maps Φ^{L} preserve the set of ECH classes *η* with *U*^{k}*η* = [∅].

More generally, *c*_{k} of an arbitrary symplectic manifold (*X*,*ω*) is defined to be the supremum of *c*_{k}(*X*^{′},*ω*^{′}), where (*X*^{′},*ω*^{′}) is a Liouville domain that can be symplectically embedded into (*X*,*ω*).

## More Examples of ECH Capacities

The ECH capacities of a polydisk are (11)

It turns out that ECH capacities also give a sharp obstruction to symplectically embedding an ellipsoid into a polydisk. The proof uses the following analogue of Proposition 7:

Let *a*, *b*, *c*, *d* > 0 with *a*/*b* rational. (Müller) Then there is a symplectic embedding int(*E*(*a*,*b*)) → *P*(*c*,*d*) if and only if there is a symplectic embedding

As in Proposition 7, the “only if” direction in Proposition 10 follows from an explicit construction (together with [**9**]). Namely, the triangle Δ(*c* + *d*,*c* + *d*) is partitioned into a rectangle of side lengths *c* and *d* together with translates of Δ(*c*,*c*) and Δ(*d*,*d*), so there is a symplectic embedding [11]

There is a symplectic embedding int(*E*(*a*,*b*)) → *P*(*c*,*d*) if and only if *c*_{•}(*E*(*a*,*b*)) ≤ *c*_{•}(*P*(*c*,*d*)).

Copy the above proof of Theorem 1, using Proposition 10 and [**11**] in place of Proposition 7 and [**10**].

ECH capacities do not always give sharp obstructions to symplectically embedding a polydisk into an ellipsoid. For example, it is easy to check that *c*_{•}(*P*(1,1)) = *c*_{•}(*E*(1,2)). Thus ECH capacities give no obstruction to symplectically embedding *P*(1,1) into *E*(*a*,2*a*) whenever *a* > 1. However, the Ekeland–Hofer capacities (see ref. 10) show that *P*(1,1) does not symplectically embed into *E*(*a*,2*a*) whenever *a* < 3/2. And the latter bound is sharp because, according to our definitions, *P*(1,1) is a subset of *E*(3/2,3).

Theorem 9 is deduced in ref. 11 from the following more general calculation, proved using results from ref. 27. Let ‖·‖ be a norm on , regarded as a translation-invariant norm on *TT*^{2}. Let ‖·‖^{∗} denote the dual norm on *T*^{∗}*T*^{2}. Define with the canonical symplectic form on *T*^{∗}*T*^{2}.

If ‖·‖ is a norm on , then (11) [12]Here the minimum is over convex polygons Λ in with vertices in , and *P*_{Λ} denotes the closed region bounded by Λ. Also *ℓ*_{‖·‖}(Λ) denotes the length of Λ in the norm ‖·‖.

Finally, we remark that, in all known examples, the ECH capacities asymptotically recover the symplectic volume, via

Let (*X*,*ω*) be a four-dimensional Liouville domain such that *c*_{k}(*X*,*ω*) < ∞ for all *k*. Then (11)

Of course, it is the deviation of *c*_{k}(*X*,*ω*)^{2}/*k* from 4 vol(*X*,*ω*) that gives rise to nontrivial symplectic embedding obstructions.

## Acknowledgments

The author thanks Paul Biran and Dusa McDuff for patiently explaining the ball packing and ellipsoid embedding stories, and Dorothee Müller for explaining Proposition 10 and its proof. This work was partially supported by National Science Foundation Grant DMS-0806037.

## Footnotes

- ↵
^{1}E-mail: hutching{at}math.berkeley.edu.

Author contributions: M.H. wrote the paper.

The author declares no conflict of interest.

This article is a PNAS Direct Submission.

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