Beyond ECH capacities
Beyond ECH capacities
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DOI：
10.2140/gt.2016.20.1085
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发表时间：
2016-01-01
影响因子：
2
通讯作者：
Hutchings, Michael
中科院分区：
文献类型：
--
作者：
Hutchings, Michael
ECH(Embedded Contact Homology)能力给将一个有边界的四维辛流形辛嵌入到另一个四维辛流形上提供了障碍。众所周知，当域和目标是椭球体时(由McDuff证明)，以及更一般地当域是“凹环形域”而目标是“凸环形域”(由Cristofaro-Gardiner证明)时，这些障碍是尖锐的。然而，ECH容量通常不会给出尖锐的障碍，例如在许多情况下当域是多磁盘时。本文利用ECH中更精细的信息，给出了当区域为多圆域或更一般的凸圆环域时，更强的辛嵌入障碍。我们利用这些新的障碍来反驳Hind和Lisi关于多圆盘到球的辛嵌入的一个结果，并将其推广到阻挡多圆盘到椭球体的某些辛嵌入。我们还得到了将一个多圆盘辛嵌入另一个多圆盘的一个新的障碍，特别是证明了Schlenk猜想的一个四维情形。
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 embedding obstructions when the domain is a polydisk, or more generally a convex toric domain. We use these new obstructions to reprove a result of Hind and Lisi on symplectic embeddings of a polydisk into a ball, and generalize this to obstruct some symplectic embeddings of a polydisk into an ellipsoid. We also obtain a new obstruction to symplectically embedding one polydisk into another, in particular proving the four-dimensional case of a conjecture of Schlenk.