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 is related to previously known criteria for when a disjoint union of balls can be symplectically embedded into a ball. The new theory of "ECH capacities" gives general obstructions to symplectic embeddings in four dimensions which turn out to be sharp in the above cases.
We prove in this paper that any 4-dimensional symplectic manifold is essentially made of finitely many symplectic ellipsoids. The key tool is a singular analogue of Donaldson's symplectic hypersurfaces in irrational symplectic manifolds.
We construct symplectic embeddings of ellipsoids of dimension $2n \ge 6$ into the product of a 4-ball or 4-dimensional cube with Euclidean space. A sequence of these embeddings can be shown to be optimal.
A stabilized polydisc is a product of a symplectic polydisc and several copies of the complex plane. This paper gives a complete characterization of symplectic embeddings of stabilized polydiscs into other stabilized polydiscs.
An embedding $φ\colon (M_1, ω_1) \to (M_2, ω_2)$ (of symplectic manifolds of the same dimension) is called $ε$-symplectic if the difference $φ^* ω_2 - ω_1$ is $ε$-small with respect to a fixed Riemannian metric on $M_1$. We prove that if a sequence of $ε$-symplectic embeddings converges uniformly (on compact subsets) to another embedding, then the limit is $E$-symplectic, where the number $E$ depends only on $ε$ and $E (ε) \to 0$ as $ε\to 0$. This generalizes $C^0$-rigidity of symplectic embeddings, and answers a question in topological quantum computing by Michael Freedman. As in the symplect…
Given two 4-dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a particular case of a more general result. Given two convex toric domains whose first and second ECH capacities satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two convex toric domains is noncontractible. We show how the constructed loops become contractible if the target domain becomes large enough. The proof i…
Given two $2n$--dimensional symplectic ellipsoids whose symplectic sizes satisfy certain inequalities, we show that a certain map from the $n$--torus to the space of symplectic embeddings from one ellipsoid to the other induces an injective map on singular homology with mod $2$ coefficients. The proof uses parametrized moduli spaces of $J$--holomorphic cylinders in completed symplectic cobordisms.
A complete embedding is a symplectic embedding $ι:Y\to M$ of a geometrically bounded symplectic manifold $Y$ into another geometrically bounded symplectic manifold $M$ of the same dimension. When $Y$ satisfies an additional finiteness hypothesis, we prove that the truncated relative symplectic cohomology of a compact subset $K$ inside $Y$ is naturally isomorphic to that of its image $ι(K)$ inside $M$. Under the assumption that the torsion exponents of $K$ are bounded we deduce the same result for relative symplectic cohomology. We introduce a technique for constructing complete embeddings usin…
In this paper we study symplectic embedding questions for the $\ell_p$-sum of two discs in ${\mathbb R}^4$, when $1 \leq p \leq \infty$. In particular, we compute the symplectic inner and outer radii in these cases, and show how different kinds of embedding rigidity and flexibility phenomena appear as a function of the parameter $p$.
In this paper we establish new restrictions on symplectic embeddings of certain convex domains into symplectic vector spaces. These restrictions are stronger than those implied by the Ekeland-Hofer capacities. By refining an embedding technique due to Guth, we also show that they are sharp.
This note constructs sharp obstructions for stabilized symplectic embeddings of an ellipsoid into a ball, in the case when the initial four-dimensional ellipsoid has `eccentricity' of the form 3n-1 for some integer n.
McDuff and Schlenk have recently determined exactly when a four-dimensional symplectic ellipsoid symplectically embeds into a symplectic ball. Similarly, Frenkel and Müller have recently determined exactly when a symplectic ellipsoid symplectically embeds into a symplectic cube. Symplectic embeddings of more complicated structures, however, remain mostly unexplored. We study when a symplectic ellipsoid $E(a,b)$ symplectically embeds into a polydisc $P(c,d)$. We prove that there exists a constant $C$ depending only on $d/c$ (here, $d$ is assumed greater than $c$) such that if $b/a$ is greater t…
We prove packing stability for any closed symplectic manifold with rational cohomology class. This will rely on a general symplectic embedding result for ellipsoids which assumes only that there is no volume obstruction and that the domain is sufficiently thin relative to the target. We also obtain easily computable bounds for the Embedded Contact Homology capacities which are sufficient to imply the existence of some volume preserving embeddings in dimension 4.
We show that the cylinder Z^{2n}(1):= B^2(1)\times \mathbb{R}^{2(n-1)} embeds symplectically into B^4(R) \times \mathbb{R}^{2(n-2)} if R \geq \sqrt{3}.
McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a four-dimensional ball, and found that when the ellipsoid is close to round, the answer is given by an "infinite staircase" determined by the odd-index Fibonacci numbers. We show that this result still holds in higher dimensions when we "stabilize" the embedding problem.
The rational homology balls $B_n$ appeared in Fintushel and Stern's rational blow-down construction [FS2]. Later, Symington [Sy1], defined this operation in the symplectic category. In [Kh2], the author defined the inverse procedure, the symplectic rational blow-up. In this paper, we study the obstructions to symplectically rationally blowing up a symplectic 4-manifold, i.e. the obstructions to symplectically embedding the rational homology balls $B_n$ into a symplectic 4-manifold. We prove a theorem and give additional examples which suggest that in order to symplectically embed the rational …
This note describes a correct way to perform the inflation procedures claimed in the papers on embedding ellipsoids, Journ. Top. 2 (2009), 1-22 and 589-623. The idea is to inflate along a collection of transversally and positively intersecting symplectic surfaces (as in Li--Usher (JSG 2006)) rather than along a single embedded surface.
Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. In "Symplectic embeddings into four-dimensional concave toric domains", the author, Choi, Frenkel, Hutchings and Ramos computed the ECH capacities of all "concave toric domains", and showed that these give sharp obstructions in several interesting cases. We show that these obstructions are sharp for all symplectic embeddings of concave toric domains into "convex" ones. In an appendix with Choi, we prove a new formula for the ECH capacities of convex toric domains, which …
If P is a polydisk with radii R_1 < ... < R_n and P' is a polydisk with radii R'_1 < ... < R'_n, then we construct a symplectic embedding from P into P' provided that C(n) R_1 < R'_1 and C(n) R_1 ... R_n < C(n) R'_1 ... R'_n. Up to a constant factor, these conditions are optimal.
In this paper we obtain sharp obstructions to the symplectic embedding of the lagrangian bidisk into four-dimensional balls, ellipsoids and symplectic polydisks. We prove, in fact, that the interior of the lagrangian bidisk is symplectomorphic to a concave toric domain using ideas that come from billiards on a round disk. In particular, we answer a question of Ostrover. We also obtain sharp obstructions to some embeddings of ellipsoids into the lagrangian bidisk.
I prove that the open unit cube can be symplectically embedded into a longer polydisc in such a way that the area of each section satisfies a sharp bound and the complement of each section is path-connected. This answers a variant of a question by F. Schlenk.
We present a novel $C^0$-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by J.-C. Sikorav and Y. Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of $C^0$-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from $J$-holomorphic curv…
ECH capacities give obstructions to symplectically embedding one symplectic four-manifold with boundary into another. We compute the ECH capacities of a large family of symplectic four-manifolds with boundary, called "concave toric domains". Examples include the (nondisjoint) union of two ellipsoids in $\mathbb{R}^4$. We use these calculations to find sharp obstructions to certain symplectic embeddings involving concave toric domains. For example: (1) we calculate the Gromov width of every concave toric domain; (2) we show that many inclusions of an ellipsoid into the union of an ellipsoid and…
We extend the ``Extension after Restriction Principle'' for symplectic embeddings of bounded starlike domains to a large class of symplectic embeddings of unbounded starlike domains.
We prove here a quantitative $h$-principle statement that applies to isotropic embeddings of discs. We then apply it to get $C^0$-flexibility and rigidity results in symplectic geometry. On the flexible side, we prove that a symplectic homeomorphism might take a symplectic disc to a smooth isotropic one. We also get a $C^0$-rigidity result for the action of a symplectic homeomorphism on the reduction of a coisotropic submanifold.
We show that many toric domains $X$ in $R^4$ admit symplectic embeddings $ϕ$ into dilates of themselves which are knotted in the strong sense that there is no symplectomorphism of the target that takes $ϕ(X)$ to $X$. For instance $X$ can be taken equal to a polydisk $P(1,1)$, or to any convex toric domain that both is contained in $P(1,1)$ and properly contains a ball $B^4(1)$; by contrast a result of McDuff shows that $B^4(1)$ (or indeed any four-dimensional ellipsoid) cannot have this property. The embeddings are constructed based on recent advances on symplectic embeddings of ellipsoids, th…
In this paper, we compute the embedded contact homology (ECH) capacities of the disk cotangent bundles $D^*S^2$ and $D^*\mathbb{R} P^2$. We also find sharp symplectic embeddings into these domains. In particular, we compute their Gromov widths. In order to do that, we explicitly calculate the ECH chain complexes of $S^*S^2$ and $S^* \mathbb{R} P^2$ using a direct limit argument on the action inspired by Bourgeois's Morse-Bott approach and ideas from Nelson-Weiler's work on the ECH of prequantization bundles. Moreover, we use integrable systems techniques to find explicit symplectic embeddings.…
In this paper we obtain new obstructions to symplectic embeddings of the four-dimensional polydisk $P(a,1)$ into the ball $B(c)$ for $2\leq a<\frac{\sqrt{7}-1} {\sqrt{7}-2} \approx 2.549$, extending work done by Hind-Lisi and Hutchings. Schlenk's folding construction permits us to conclude our bound on $c$ is optimal. Our proof makes use of the combinatorial criterion necessary for one "convex toric domain" to symplectically embed into another introduced by Hutchings in \cite{Beyond}. Additionally, we prove that if certain symplectic embeddings of four dimensional convex toric domains exist th…
In this note we show that one open four dimensional ellipsoid embeds symplectically into another if and only the ECH capacities of the first are no larger than those of the second. This proves a conjecture due to Hofer. The argument uses the equivalence of the ellipsoidal embedding problem with a ball embedding problem that was recently established by McDuff. Its method is inspired by Hutchings' recent results on embedded contact homology (ECH) capacities but does not use them.
We study symplectic embeddings of ellipsoids into balls. In the main construction, we show that a given embedding of 2m-dimensional ellipsoids can be suspended to embeddings of ellipsoids in any higher dimension. In dimension 6,s if the ratio of the areas of any two axes is sufficiently large then the ellipsoid is flexible in the sense that it fully fills a ball. We also show that the same property holds in all dimensions for sufficiently thin ellipsoids E(1,..., a). A consequence of our study is that in arbitrary dimension a ball can be fully filled by any sufficiently large number of identic…
In this note, we obtain new obstructions to symplectic embeddings of a product of disks (a polydisk) into a 4-dimensional ball. The polydisk P(r,s) is the product of the disk of area r with the disk of area s. The ball of capacity a, denoted B(a), is the ball with πr^2 \le a. We show P(1,2) embeds in B^4(a) if and only if a is at least 3. This shows the inclusion of P(1,2) in B^4(3) is optimal. The necessity of a \ge 3 implies that for this particular embedding problem neither the Ekeland-Hofer nor ECH capacities give a sharp obstruction. We contrast this with the case of ellipsoid embeddings …
We show how to reduce the problem of symplectically embedding one 4-dimensional rational ellipsoid into another to a problem of embedding disjoint unions of balls into appropriate blow ups of \C P^2. For example, the problem of embedding the ellipsoid E(1,k) into a ball B is equivalent to that of embedding k disjoint equal balls into \C P^2, and so can be solved by the work of Gromov, McDuff--Polterovich and Biran. (Here k is the ratio of the area of the major axis to that of the minor axis.) As a consequence we show that the ball may be fully filled by the ellipsoid E(1,k) for k=1,4 and all k…
The Rabinowitz-Floer homology groups $RFH_*(M,W)$ are associated to an exact embedding of a contact manifold $(M,ξ)$ into a symplectic manifold $(W,ω)$. They depend only on the bounded component $V$ of $W\setminus M$. We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$, which in turn maps to Rabinowitz-Floer homology $RFH_*(M,W)$, which then maps to symplectic cohomology of $V$. We compute $RFH_*(ST^*L,T^*L)$, where $ST^*L$ is the unit cosphere bundle of a closed manifold $L$. As an application, we prove that the image of an exact contact…
In determining when a four-dimensional ellipsoid can be symplectically embedded into a ball, McDuff and Schlenk found an infinite sequence of "ghost" obstructions that generate an infinite "ghost staircase" determined by the even index Fibonacci numbers. The ghost obstructions are not visible for the four-dimensional embedding problem because strictly stronger obstructions also exist. We show that in contrast, the embedding constraints associated to the ghost obstructions are sharp for the stabilized problem; moreover, the corresponding optimal embeddings are given by symplectic folding. The p…
In this article we study proper symplectic and iso-symplectic embeddings of $4$--manifolds in $6$--manifolds. We show that a closed orientable smooth $4$--manifold admitting a Lefschetz fibration over $\C P^1$ admits a symplectic embedding in the symplectic manifold $(\C P^1 \times \C P^1 \times \C P^1, ω_{pr}),$ where $ω_{pr}$ is the product symplectic form on $\C P^1 \times \C P^1 \times \C P^1.$ We also show that there exists a sub-critical Weinstein $6$--manifold in which all finite type Weinstein $4$--manifolds admit iso-symplectic embeddings.
We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. Applications include the proof of Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic space, the construction of a new symplectic capacity, obstructions to Lagrangian embeddings into uniruled symplectic manifolds, a quantitative version of Arnold's chord conjecture, and estimates on the size of Weinstein neighbourhoods. The main technical ingredient is transversality for the relevant mod…
The third named author has been developing a theory of "higher" symplectic capacities. These capacities are invariant under taking products, and so are well-suited for studying the stabilized embedding problem. The aim of this note is to apply this theory, assuming its expected properties, to solve the stabilized embedding problem for integral ellipsoids, when the eccentricity of the domain has the opposite parity of the eccentricity of the target and the target is not a ball. For the other parity, the embedding we construct is definitely not always optimal; also, in the ball case, our methods…
We obtain universal models for several types of locally conformal symplectic manifolds via pullback or reduction. The relation with recent embedding results for locally conformal Kähler manifolds is discussed.
Recently, McDuff and Schlenk determined the function c_{EB}(a) whose value at a is the infimum of the size of a 4-ball into which the ellipsoid E(1,a) symplectically embeds (here, a >= 1 is the ratio of the area of the large axis to that of the smaller axis of the ellipsoid). In this paper we look at embeddings into four-dimensional cubes instead, and determine the function c_{EC}(a) whose value at a is the infimum of the size of a 4-cube C^{4}(A) = D^{2}(A) times D^{2}(A) into which the ellipsoid E(1,a) symplectically embeds (where D^{2}(A) denotes the disc in mathbb{R}^{2} of area A). As in …
In previous work, the second author and Müller determined the function $c(a)$ giving the smallest dilate of the polydisc $P(1,1)$ into which the ellipsoid $E(1,a)$ symplectically embeds. We determine the function of two variables $c_b(a)$ giving the smallest dilate of the polydisc $P(1,b)$ into which the ellipsoid $E(1,a)$ symplectically embeds for all integers $b \geqslant 2$. It is known that for fixed $b$, if $a$ is sufficiently large then all obstructions to the embedding problem vanish except for the volume obstruction. We find that there is another kind of change of structure that appear…
ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain and target are ellipsoids (proved by McDuff), and more generally when the domain is a "concave toric domain" and the target is a "convex toric domain" (proved by Cristofaro-Gardiner). However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk. This paper uses more refined information from ECH to give stronger symplectic embed…
In any dimension $2n \ge 6$ we show that certain spaces of symplectic embeddings of a polydisk into a product $B^4 \times \Bbb R^{2(n-2)}$ of a $4$-ball and Euclidean space, are not path connected. We also show that any pair of such nonisotopic embeddings can never be extended to the same ellipsoid.
We introduce the notion of a symplectic capacity relative to a coisotropic submanifold of a symplectic manifold, and we construct two examples of such capacities through modifications of the Hofer-Zehnder capacity. As a consequence, we obtain a non-squeezing theorem for symplectic embeddings relative to coisotropic constraints and existence results for leafwise chords on energy surfaces.
We prove the following middle-dimensional non-squeezing result for analytic symplectic embeddings of domains in $\mathbb{R}^{2n}$. Let $φ: D \hookrightarrow \mathbb{R}^{2n}$ be an analytic symplectic embedding of a domain $D \subset \mathbb{R}^{2n}$ and $P$ be a symplectic projector onto a linear $2k$-dimensional symplectic subspace $V\subset \mathbb{R}^{2n}$. Then there exists a positive function $r_0:D\rightarrow (0,+ \infty)$, bounded away from $0$ on compact subsets $K \subset D$, such that the inequality $Vol_{2k}(Pφ(B_r(x)),ω^k _{0|V})\geq π^{k} r^{2k}$ holds for every $x \in D$ and for …
We compute the homotopy type of the space of T^n-equivariant symplectic embeddings from the standard 2n-dimensional ball of some fixed radius into a 2n-dimensional symplectic-toric manifold M, and use this computation to define a Z-valued step function on the positive real line which is an invariant of the symplectic-toric type of M. We conclude with a discussion of the partially equivariant case of this result.
Quite recently, McDuff showed that the existence of a symplectic embedding of one four-dimensional ellipsoid into another can be established by comparing their corresponding sequences of ECH capacities. In this note we show that these sequences can be encoded in a generating function, which gives several new equivalent formulations of McDuff's theorem.
We show that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that the homology cobordism group is generated by Stein fillable 3-manifolds. We also find obstructions on smooth embeddings: there exists 3-manifolds which cannot smoothly embed in a way that appropriately respect orientations in any symplectic manifold with weakly convex boundary. This embedding obstruction can also be used to detect exotic smooth structures on 4…
As has been known since the time of Gromov's Nonsqueezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry. After surveying some of the most important ways of measuring the size of a symplectic set, these notes discuss some recent developments concerning the question of when a 4-dimensional ellipsoid can be symplectically embedded in a ball. This problem turns out to have unexpected relations to the properties of continued fractions and of exceptional curves in blow ups of the complex projective plane. It is also related to questions of lattice packing of planar …
