Dynamically convex and global surface of section in $L(p,p-1)$ from the viewpoint of ECH

Abstract

The notion of dynamically convex has been studied since it was introduced by Hofer, Wysocki and Zehnder. In particular, they showed that there muse exist a global surface of section of disk type binding a periodic orbit for the Reeb vector field in dynamically convex $(S^{3},\lambda)$ by using pseudoholomorphic curves. Recently Hryniewicz and Salom\~ao showed the same result for $L(2,1)$ by developing the original technique and after that Schneider generalized it to $(L(p,1),\xi_{\mathrm{std}})$. The main purpose of this paper is to introduce an alternative approach of using Embedded contact homology. In particular, we find a global surface of section of disk type in dynamically convex $(L(p,p-1),\xi_{\mathrm{std}})$ and relate their periods to the first ECH spectrum.