ECH capacities and the Ruelle invariant
ECH capacities and the Ruelle invariant
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DOI：
10.1007/s11784-022-00968-3
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发表时间：
2022-06-01
影响因子：
1.8
通讯作者：
Hutchings, Michael
中科院分区：
文献类型：
--
作者：
Hutchings, Michael
ECH容量是一个与任意辛四流形相关联的真实的数列，它关于辛嵌入是单调的。对于R-4中的紧致星型区域，ECH容量渐近恢复区域的体积。我们猜想，一个启发式的参数，一般的误差项在这个渐近公式收敛到一个常数确定的“Ruelle不变量”的措施，平均旋转的Reeb流的边界上。我们的主要结果是证明了这一猜想的一大类环面域。作为推论，我们得到了具有相同体积的开环面域的辛嵌入的一般性障碍。对于R-4中更一般的域，我们将误差项与以前已知的指数从2/5改进到1/4。
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 corollary, we obtain a general obstruction to symplectic embeddings of open toric domains with the same volume. For more general domains in R-4, we bound the error term with an improvement on the previously known exponent from 2/5 to 1/4.