关于凸域间辛嵌入的障碍
Obstruction to Symplectic Embeddings between Toric Domains
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中文总结 AI 辅助
该研究利用嵌入接触同调，扩展Hutchings准则得到三类凸域辛嵌入的障碍，经应用验证其部分障碍强于ECH容量导出的障碍。
中文摘要 AI 辅助
我们利用嵌入接触同调（ECH）研究四维凸域间的辛嵌入。扩展Hutchings关于凸凸域间嵌入的准则，我们得到三类嵌入的障碍：从凸凸域到凹凸域、从半弱凸凸域到凸凸域、以及从凹凸域到凹凸域。这些障碍通过凸、凹、半弱凸生成元的因式分解，以及关联其ECH指标与作用量的组合约束来表述。作为应用，我们重新得到多圆盘到柱体并集的辛嵌入的尖锐障碍，并获得某些四边形凸域到球与椭球嵌入的尖锐结果。我们还证明，这些障碍在部分情形下严格强于仅来自ECH容量的障碍。
英文摘要
We study symplectic embeddings between four-dimensional toric domains using embedded contact homology. Extending Hutchings' criterion for embeddings between convex toric domains, we obtain obstructions for embeddings from convex toric domains into concave toric domains, from semi-weakly convex toric domains into convex toric domains, and from concave toric domains into concave toric domains. The obstructions are formulated in terms of factorizations of convex, concave, and semi-weakly convex generators together with combinatorial constraints relating their ECH indices and actions. As applications, we recover the sharp obstruction for symplectic embeddings of a polydisk into a union of cylinders, and obtain sharp results for embeddings of certain quadrilateral toric domains into balls and ellipsoids. We also show that these obstructions are in some cases strictly stronger than those coming only from ECH capacities.
