Large Steklov eigenvalues on hyperbolic surfaces

In this paper, we first construct a sequence of hyperbolic surfaces with connected geodesic boundary such that the first normalized Steklov eigenvalue ${\tilde{\sigma }}_1$ tends to infinity. We then prove that as $g\to \mathrm{\infty }$, a generic $\mathrm{\Sigma }\in {\mathcal{M}}_{g,n}(L_g)$ satisfies ${\tilde{\sigma }}_1(\mathrm{\Sigma })>C\cdot \mathrm{\parallel }{L}_g{\mathrm{\parallel }}_1$ 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 $\mathrm{\parallel }{L}_g{\mathrm{\parallel }}_1\to \mathrm{\infty }$ satisfies certain conditions.