Large Steklov eigenvalues on hyperbolic surfaces

Abstract

In this paper, we first construct a sequence of hyperbolic surfaces with connected geodesic boundary such that the first normalized Steklov eigenvalue $\tilde{\ddot{I}}{ƒ}_1$ tends to infinity. We then prove that as $g\to \infty$, a generic $\hat{I}\pounds \in {\mathcal{M}}_{g,n}(L_g)$ satisfies $\tilde{\ddot{I}}{ƒ}_1(\hat{I}\pounds )>C\cdot \Vert L_g{\Vert }_1$ where $C$ is a positive universal constant. Here ${\mathcal{M}}_{g,n}(L_g)$ is the moduli space of hyperbolic surfaces of genus $g$ and $n$ boundary components of length $L_g=(L_g^1,\cdots ,L_g^n)$ endowed with the Weil-Petersson metric where $\Vert L_g{\Vert }_1\to \infty$ satisfies certain conditions.

20pages, new results added, second theorem is improved