Elliptic bindings and the first ECH spectrum for convex Reeb flows on lens spaces

Abstract

In this paper, at first we introduce a sufficient condition for a rational unknotted Reeb orbit $\gamma$ in a lens space to be elliptic by using the rational self-linking number $sl_{\xi}^{\mathbb{Q}}(\gamma)$ and the Conley-Zehnder index $\mu_{\mathrm{disk}}(\gamma^{p})$, where $\mu_{\mathrm{disk}}$ is the Conley-Zehnder index with respect to a trivialization induced by a binding disk. As a consequence, we show that a periodic orbit $\gamma$ in dynamically convex $L(p,1)$ must be elliptic if $\gamma^{p}$ binds a Birkhoff section of disk type and has $\mu_{\mathrm{disk}}(\gamma^{p})=3$. It was proven in \cite{Sch} that such an orbit always exists in a dynamically convex $L(p,1)$. Next, we estimate the first ECH spectrum on dynamically convex $L(3,1)$. In particular, we show that the first ECH spectrum on a strictly convex (or non-degenerate dynamically convex) $(L(3,1),\lambda)$ is equal to the infimum of contact areas of certain Birkhoff sections of disk type.The key of the argument is to conduct technical computations regarding indices present in ECH and to observe the topological properties of rational open book decompositions supporting $(L(3,1),\xi_{\mathrm{std}})$ coming from $J$-holomorphic curves.