An elementary alternative to ECH capacities.
An elementary alternative to ECH capacities.
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DOI：
10.1073/pnas.2203090119
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发表时间：
2022-08-30
影响因子：
11.1
通讯作者：
Hutchings, Michael
中科院分区：
文献类型：
--
作者：
Hutchings, Michael
辛几何是经典力学的基础几何。 20 世纪 80 年代的格罗莫夫非挤压定理表明，确定相空间中的一个域何时可以嵌入到另一个域中同时保留辛结构是一个微妙的问题。从那时起，各种“辛能力”被开发出来来研究这个问题。特别是，嵌入式接触同源（ECH）能力有时会在四维情况下给出尖锐的结果。本文介绍了一系列辛容量，其功率与 ECH 容量大致相同，但以更基本的方式定义。这种结构的变体预计将在理解辛嵌入方面取得进一步进展。嵌入接触同调（ECH）能力是辛四流形的数值不变量序列，它给辛嵌入带来（有时是尖锐的）障碍。这些能力是使用嵌入的接触同源性来定义的，并且目前建立它们的基本属性需要 Seiberg-Witten 理论。在本文中，我们仅使用全纯曲线的基本概念定义了四个维度的辛容量序列。这些容量满足与 ECH 容量相同的基本属性，并且与已计算后者的主要示例的 ECH 容量一致，即凸域和凹域。这些功能对于阻止辛嵌入到闭合辛四流形中也很有用。这项工作的灵感来自 McDuff 和 Siegel 最近的预印本 [D. McDuff, K. Siegel, arXiv [预印本] (2021)]，从有理辛场论 (SFT) 中给出了辛容量的类似基本替代方案。
Symplectic geometry is the basic geometry underlying classical mechanics. The Gromov nonsqueezing theorem from the 1980s showed that it is a subtle problem to determine when one domain in phase space can be embedded into another while preserving the symplectic structure. Since then, various “symplectic capacities” have been developed to study this question. In particular, the embedded contact homology (ECH) capacities give sometimes sharp results in the four-dimensional case. This article introduces a sequence of symplectic capacities that have roughly the same power as the ECH capacities, but are defined in a more elementary way. Variants of this construction are expected to lead to further progress on understanding symplectic embeddings. The embedded contact homology (ECH) capacities are a sequence of numerical invariants of symplectic four-manifolds that 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 paper we define a sequence of symplectic capacities in four dimensions using only basic notions of holomorphic curves. The 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, namely convex and concave toric domains. The capacities are also useful for obstructing symplectic embeddings into closed symplectic four-manifolds. This work is inspired by a recent preprint of McDuff and Siegel [D. McDuff, K. Siegel, arXiv [Preprint] (2021)], giving a similar elementary alternative to symplectic capacities from rational symplectic field theory (SFT).
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