Title: ECH capacities and the Ruelle invariant Show Chinese title

Title: ECH容量和Ruelle不变量

Abstract: 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. Show Chinese abstract

Abstract: ECH容量是与任何辛四维流形相关的一系列实数，它们在辛嵌入下是单调的。 已知对于R^4中的紧致星形区域，ECH容量渐近地恢复该区域的体积。 我们通过一个启发式论证提出一个猜想，即这个渐近公式中的误差项通常收敛到由“Ruelle不变量”决定的常数，该不变量衡量边界上Reeb流动的平均旋转。 我们的主要结果是对一大类环状域证明了这一猜想。 作为推论，我们得到了一个关于具有相同体积的开环状域的辛嵌入的一般障碍。 对于R^4中的更一般的区域，我们将误差项进行了限制，并将之前已知的指数从2/5改进到1/4。