The ECH capacities for the rotating Kepler problem

ECH capacities and the Ruelle invariant

The ECH capacities are a sequence of real numbers associated to any symplectic four-manifold, which are monotone with respect to symplectic embeddings. It is known that for a compact star-shaped domain in R^4, the ECH capacities asymptotically recover the volume of the domain. We conjecture, with a heuristic argument, that generically the error term in this asymptotic formula converges to a constant determined by a "Ruelle invariant" which measures the average rotation of the Reeb flow on the boundary. Our main result is a proof of this conjecture for a large class of toric domains. As a corol…

Beyond ECH capacities

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…

Towers of Looijenga pairs and asymptotics of ECH capacities

ECH capacities are rich obstructions to symplectic embeddings in 4-dimensions that have also been seen to arise in the context of algebraic positivity for (possibly singular) projective surfaces. We extend this connection to relate general convex toric domains on the symplectic side with towers of polarised toric surfaces on the algebraic side, and then use this perspective to show that the sub-leading asymptotics of ECH capacities for all convex and concave toric domains are $O(1)$. We obtain sufficient criteria for when the sub-leading asymptotics converge in this context, generalising resul…

定量包埋接触同源性

Quantitative embedded contact homology

定义“Liouville域”为具有接触型边界的紧精确辛流形。我们使用嵌入接触同调分配给每个四维Liouville域（或其子集）的一个序列的实数，我们称之为“ECH容量”。刘维域的ECH能力定义在其边界的“ECH谱”，它衡量的数量辛行动需要代表某些类嵌入接触的同源。使用协边映射嵌入接触的同源性（定义在联合工作与Taubes），我们表明，ECH的能力是单调的辛嵌入。我们计算的ECH能力的椭球，polydisks，某些子集的余切束T2，和不相交的工会的例子，其中ECH能力是已知的。由此产生的辛嵌入障碍是尖锐的，在一些有趣的情况下，例如问题的椭球嵌入到一个球（如工作所示的麦克达夫-施伦克）或嵌入一个不相交的联盟的球到一个球。我们还提出了一个猜想，根据该猜想，刘维域的ECH能力的渐近恢复其辛体积的状态和证据。

凹奇异环面域的 ECH 容量

ECH capacities of concave singular toric domains

本文推广四维环面域至边界为透镜空间 L(n,1) 的奇异环面域，并计算一类凹奇异环面域的 ECH 容量，结果可用于研究其去奇异化空间中的嵌入问题。

Symplectic embeddings from concave toric domains into convex ones

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 …

Algebraic capacities

We study invariants coming from certain optimisation problems for nef divisors on surfaces. These optimisation problems arise in work of the author and collaborators tying obstructions to embeddings between symplectic 4-manifolds to questions of positivity for (possibly singular) algebraic surfaces. We develop the general framework for these invariants and prove foundational results on their structure and asymptotics. We describe the connections these invariants have to Embedded Contact Homology (ECH) and the Ruelle invariant in symplectic geometry, and to min-max widths in the study of minima…

旋转开普勒问题的特殊凹环面域

The special concave toric domain for the rotating Kepler problem

针对能量≤-3/2的旋转开普勒问题（RKP），通过两种辛正则化识别相空间有界分量，构造特殊凹环面域（SCTD）及组合树结构，推进辛嵌入理解并提供哈密顿动力学研究新工具。

关于嵌入辛椭球的Hessel猜想

The Hofer conjecture on embedding symplectic ellipsoids

本文证明了一个开的四维椭球辛嵌入另一个开的四维椭球当且仅当第一个开的四维椭球的ECH容量不大于第二个开的四维椭球。这证明了一个猜想，由于哈塞尔。该论点使用的等效椭球嵌入问题与球嵌入问题，最近建立的麦克达夫。其方法的灵感来自Hutchings最近的嵌入式接触同源性（ECH）容量的结果，但没有使用它们。

关于极小极大 Seiberg-Witten Floer 生成元能量的一个估计

An estimate on energy of min-max Seiberg-Witten Floer generators

本文改进极小极大 Seiberg-Witten Floer 生成元的能量估计，从而无需辛 4 维流形中的嵌入球即可直接证明 ECH 容量恢复体积，并进一步估计收敛速度。

四维凸环面域辛嵌入问题中的障碍

Obstructions in Symplectic Embedding Problems for Four-Dimensional Convex Toric Domains

本文推广Hutchings的ECH方法，研究四维凸环面域及其并集的辛嵌入障碍，并利用交正性发现新障碍，阻碍嵌入间同伦的存在。

Symplectic embeddings into disk cotangent bundles

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.…