Shiri Artstein-Avidan, Vitali D. Milman, Yaron Ostrover
In this work we bring together tools and ideology from two different fields, Symplectic Geometry and Asymptotic Geometric Analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity of a convex body by its volume radius.
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We use positive S^1-equivariant symplectic homology to define a sequence of symplectic capacities c_k for star-shaped domains in R^{2n}. These capacities are conjecturally equal to the Ekeland-Hofer capacities, but they satisfy axioms which allow them to be computed in many more examples. In particular, we give combinatorial formulas for the capacities c_k of any "convex toric domain" or "concave toric domain". As an application, we determine optimal symplectic embeddings of a cube into any convex or concave toric domain. We also extend the capacities c_k to functions of Liouville domains whic…
Characterizing symplectic capacities on ellipsoids
本文研究与椭球上第 k 个 Ekeland-Hofer 容量相等能否刻画辛容量，证明该条件在二维凸 toric 域成立，但在更高维及部分单调情形失败。
We give the first concrete examples of symplectic capacities that are not target-representable. This provides some answers to a question by Cieliebak, Hofer, Latschev, and Schlenk.
In this paper we give concrete estimations for the pseudo symplectic capacities of toric manifolds in combinatorial data. Some examples are given to show that our estimates can compute their pseudo symplectic capacities. As applications we also estimate the symplectic capacities of the polygon spaces. Other related results are impacts of symplectic blow-up on symplectic capacities, symplectic packings in symplectic toric manifolds, the Seshadri constant of an ample line bundle on toric manifolds, and symplectic capacities of symplectic manifolds with $S^1$-action.
We derive new estimates for the Gromov width of certain domains in C^2.
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.
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…
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.
Symplectic capacities, unperturbed curves, and convex toric domains
本文利用无虚拟扰动的显式伪全纯曲线技术定义一系列辛容量，并计算所有四维凸环面域的对应容量，从而为稳定辛嵌入问题提供新的、有时尖锐的障碍。
$S^1$-equivariant relative symplectic cohomology and relative symplectic capacities
本文构造了$S^1$-等变相对辛上同调，并由此建立相对Gutt-Hutchings容量与相对辛（上）同调容量，证明它们能探测紧子集的可移位性和沉重性，且在凸性假设下前两者相等。
The ECH capacities are a sequence of numerical invariants of symplectic four-manifolds which give (sometimes sharp) obstructions to symplectic embeddings. These capacities are defined using embedded contact homology, and establishing their basic properties currently requires Seiberg-Witten theory. In this note we define a new sequence of symplectic capacities in four dimensions using only basic notions of holomorphic curves. The new capacities satisfy the same basic properties as ECH capacities and agree with the ECH capacities for the main examples for which the latter have been computed, nam…
The ECH and alternative ECH capacities of closed symplectic 4-manifolds
该研究证明了具有理辛形式的闭辛4维流形的ECH容量无穷，给出替代ECH容量无穷的辛4维流形首批例子，并验证了对应ECH Weyl律的有界次渐近行为。
In this paper we prove that the symplectic Brunn-Minkowski inequality, established by Artstein-Avidan and Ostrover for the Ekeland-Hofer-Zehnder capacity, fails to hold for the Gutt-Hutchings capacities of index greater than one.
We construct an open bounded star-shaped set in R^4 whose cylindrical capacity is strictly bigger than its proper displacement energy.
For a convex domain in the standard Euclidean symplectic space which is invariant under a linear anti-symplectic involution $τ$ we show that its Ekeland-Hofer-Zehnder capacity is equal to the $τ$-symmetrical symplectic capacity of it.
A long-standing conjecture states that all normalized symplectic capacities coincide on the class of convex subsets of ${\mathbb R}^{2n}$. In this note we focus on an asymptotic (in the dimension) version of this conjecture, and show that when restricted to the class of centrally symmetric convex bodies in ${\mathbb R}^{2n}$, several symplectic capacities, including the Ekeland-Hofer-Zehnder capacity, the displacement energy capacity, and the cylindrical capacity, are all equivalent up to an absolute constant.
We prove that the generalized symplectic capacities recognize objects in symplectic categories whose objects are of the form $(M, ω)$, such that $M$ is a compact and 1-connected manifold, $ω$ is an exact symplectic form on $M$, and there exists a boundary component of $M$ with negative helicity. The set of generalized symplectic capacities is thus a complete invariant for such categories. This answers a question by Cieliebak, Hofer, Latschev, and Schlenk. It appears to be the first result concerning this question, except for recognition results for manifolds of dimension 2, ellipsoids, and pol…
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…
The spectral diameter of a symplectic ellipsoid
本研究证明辛球谱直径等于其容量，将已知界提升两倍并给出辛椭球谱直径公式，同时探究了谱直径与双球填充的关联。
We present recursive formulas which compute the recently defined "higher symplectic capacities" for all convex toric domains. In the special case of four-dimensional ellipsoids, we apply homological perturbation theory to the associated filtered L-infinity algebras and prove that the resulting structure coefficients count punctured pseudoholomorphic curves in cobordisms between ellipsoids. As sample applications, we produce new previously inaccessible obstructions for stabilized embeddings of ellipsoids and polydisks, and we give new counts of curves with tangency constraints.
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.
The shape invariant of a symplectic manifold encodes the possible area classes of embedded Lagrangian tori. Potentially this is a powerful invariant, but for most manifolds the shape is unknown. We compute the shape for 4 dimensional symplectic ellipsoids where the ratio of the factors is an integer. The full shape invariant gives stronger embedding obstructions than results by considering only monotone tori.
We introduce a new normalization condition for symplectic capacities, which we call cube normalization. This condition is satisfied by the Lagrangian capacity and the cube capacity. Our main result is an analogue of the strong Viterbo conjecture for monotone toric domains in all dimensions. Moreover, we give a family of examples where standard normalized capacities coincide but not cube normalized ones. Along the way, we give an explicit formula for the Lagrangian capacity on a large class of toric domains.
Let $M$ be a closed Fano symplectic manifold with a semifree Hamiltonian circle action with isolated maximum. We compute the Gromov width and the Hofer-Zehnder capacity of $M$ using a moment map.
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.
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.
On the agreement of symplectic capacities in high dimension
本研究将Gutt-Hutchings-Ramos关于归一化辛容量对四维单调环面域取值一致的定理推广到任意维，通过构造L形域到无穷圆柱的辛嵌入，正面解决了Gutt-Pereira-Ramos的相关猜想。
A note on the wild symplectic ellipsoids
本文通过辛2-积构造证明，任意开子集可辛同胚于包含特定Hausdorff维数集合的开集，扩展了辛几何中关于椭球体的结论。
In this work we prove a Brunn-Minkowski-type inequality in the context of symplectic geometry and discuss some of its applications.
While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. In this paper we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities. We also describe several new relations between certain symplectic capaci…
Stabilized symplectic embeddings of higher-dimensional ellipsoids
本文通过研究高维辛椭球体的拉格朗日容量，提出了一种基于弦拓扑学的方法，以确定其嵌入容量的下界。
We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions, then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic u…
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 work we discuss a conjecture of Viterbo relating the symplectic capacity of a convex body and its volume. The conjecture states that among all 2n-dimensional convex bodies with a given volume the euclidean ball has maximal symplectic capacity. We present a proof of this fact up to a logarithmic factor in the dimension, and many classes of bodies for which this holds up to a universal constant.
We establish computational results concerning the Lagrangian capacity from "Cieliebak and Mohnke - Punctured holomorphic curves and Lagrangian embeddings". More precisely, we show that the Lagrangian capacity of a 4-dimensional convex toric domain is equal to its diagonal. The proof involves comparisons between the Lagrangian capacity, the McDuff-Siegel capacities from "McDuff and Siegel - Symplectic capacities, unperturbed curves, and convex toric domains", and the Gutt-Hutchings capacities from "Gutt and Hutchings - Symplectic capacities from positive S1-equivariant symplectic homology". Wor…
Symplectic capacities of disc cotangent bundles of flat tori
本文证明了平坦Riemannian环面及Finsler环面单位圆盘余切丛上的归一化容量等于2倍systole。
The equivalence of Ekeland-Hofer and equivariant symplectic homology capacities
该研究证明了Ekeland-Hofer容量与等变辛同调容量在$\u211d^{2n}$所有星形域上一致，解决了35年的公开问题，同时建立了等变Floer同调与Morse同调的链复形同构关系。
Algebraic capacities as tropical polynomials over the reduced $c_1$-positive symplectic cone
本文用驯化辛形式与近复结构重新表述代数容量，证明其仅依赖上同调类，引入热带多项式描述容量，并给出光滑环面曲面上代数容量与ECH容量对应的纯辛证明。
Ellipsoidal superpotentials and stationary descendants
该研究计算任意维辛椭球的稳态引力派生类，推导带椭球端辛流形穿孔曲线计数的新递推公式，构建借仿射空间标准复结构显式计算的框架，并开启对作为椭球辛配边基本单元的无穷小辛配边的研究。
Remarks on symplectic capacities of $p$-products
本文研究凸区域辛 $p$-乘积的容量行为，借助张量幂技巧将 Viterbo 弱体积-容量猜想归约到高维渐近情形，并提出高阶容量猜想以排除辛球的非平凡 $p$-分解。
K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk (CHLS) posed the problem of finding a minimal generating set for the (symplectic) capacities on a given symplectic category. We show that if the category contains a certain one-parameter family of objects, then every countably Borel-generating set of (normalized) capacities has cardinality (strictly) bigger than the continuum. This appears to be the first result regarding the problem of CHLS, except for two results of D. McDuff about the category of ellipsoids in dimension 4. We also prove that every finitely differentiably generating set of …
For any Lie group $G$, we construct a $G$-equivariant analogue of symplectic capacities and give examples when $G = \mathbb{T}^k\times\mathbb{R}^{d-k}$, in which case the capacity is an invariant of integrable systems. Then we study the continuity of these capacities, using the natural topologies on the symplectic $G$-categories on which they are defined.
Comparison of symplectic capacities
本文比较了相对情形下的辛（上）同调容量与谱容量，建立了相对辛容量不等式链，提供了基于重性的相对几乎存在定理判据，并研究了辛（上）同调容量与第一个Gutt-Hutchings容量相等的宽松充分条件。
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.
A toric domain is a subset of $(\mathbb{C}^n,ω_{\text{std}})$ which is invariant under the standard rotation action of $\mathbb{T}^n$ on $\mathbb{C}^n$. For a toric domain $U$ from a certain large class for which this action is not free, we find a corresponding toric domain $V$ where the standard action is free, and for which $c(U)=c(V)$ for any symplectic capacity $c$. Michael Hutchings gives a combinatorial formula for calculating his embedded contact homology symplectic capacities for certain toric four-manifolds on which the $\mathbb{T}^2$-action is free. Our theorem allows one to extend t…
Higher symplectic capacities
本文提出基于有理辛字段理论的辛容量家族，通过L-infinity结构和局部切线约束，改进了辛嵌入障碍的计算方法。
On aspherical symplectic fillings with finite capacities of the prequantization bundles
本文研究预量子化丛的辛非球面填充问题，证明在满足特定拓扑条件且辛容量有限时，该填充必微分同胚于相应的圆盘丛。
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.
In this note we study the systoles of convex hypersurfaces in $\mathbb{R}^{2n}$ invariant under an anti-symplectic involution. We investigate a uniform upper bound of the ratio between the systole and the symmetric systole of the hypersurfaces using symplectic capacities from Floer theory. We discuss various concrete examples in which the ratio can be understood explicitly.
Symplectic capacities of $S^1$-invariant dynamically convex domains in $\mathbb{R}^4$
本文证明了在 $\mathbb{C}^2$ 中，对于任何与 Hopf 对角作用同伦的 Hamiltonian $S^1$ 作用下不变的动态凸区域，所有归一化辛容量相等，并给出了 $S^1$ 不变区域动态凸的充要条件。
In this paper we settle three basic questions concerning the Gutt-Hutchings capacities. Our primary result settles a version of the recognition question in the negative. We prove that the Gutt-Hutchings capacities together with the volume, do not constitute a complete set of symplectic invariants for star-shaped domains with smooth boundary. We also establish two independence properties. We prove that, even for star-shaped domains with smooth boundaries, these capacities are independent from the volume. We also prove that the capacities are mutually independent by constructing, for any $j \in …
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…
Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits a symplectic embedding to M. We show that the symplectic packings by ellipsoids are unobstructed for all even-dimensional tori equipped with Kahler symplectic forms and all closed hyperkahler manifolds of maximal holonomy, or, more generally, for closed Campana simple manifolds (that is, Kahler manifolds that are not unions of their complex subvarieties), as…
Inspired by the work of G. Lu on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer--Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also compute the Gromov width and the Hofer--Zehnder capacity for Cartan domains and their products.
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 …
Roughly twenty five years ago Hofer asked: can the cylinder B^2(1) \times \mathbb{R}^{2(n-1)} be symplectically embedded into B^{2(n-1)}(R) \times \mathbb{R}^2 for some R>0? We show that this is the case if R \geq \sqrt{2^{n-1}+2^{n-2}-2}. We deduce that there are no intermediate capacities, between 1-capacities, first constructed by Gromov in 1985, and n-capacities, answering another question of Hofer. In 2008, Guth reached the same conclusion under the additional hypothesis that the intermediate capacities should satisfy the exhaustion property.
McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a ball, and found that part of the answer is given by an infinite "Fibonacci staircase." Similarly, Frenkel and Müller determined when a four-dimensional ellipsoid can be symplectically embedded into the ellipsoid E(1,2) and found that part of the answer is given by a "Pell staircase." ECH capacities give an obstruction to symplectically embedding one four-dimensional ellipsoid into another, and McDuff showed that this obstruction is sharp. We use this result to give new proofs of the staircases…
Maximal Weinstein neighborhoods of symmetric R-spaces and their symplectic capacities
本文证明对称R-空间的极大Weinstein管状邻域具有稠密性，并据此计算其圆盘（余）切丛的Gromov宽度与Hofer--Zehnder容量。
We show that the spherical capacity is discontinuous on a smooth family of ellipsoidal shells. Moreover, we prove that the shell capacity is discontinuous on a family of open sets with smooth connected boundaries.
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…
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