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 \(\tilde{\sigma }_1\) ${\tilde{\sigma }}_1$ tends to infinity. We then prove that the Weil–Petersson probability that a hyperbolic surface \(\Sigma \) $\mathrm{\Sigma }$ of genus g with n boundary components of lengths \(L_g^1, \ldots , L_g^n\) $L_g^1,\dots ,L_g^n$ satisfies \(\tilde{\sigma }_1(\Sigma )>C\cdot \Vert L_g\Vert _1\) ${\tilde{\sigma }}_1(\mathrm{\Sigma })>C·{\Vert L_g\Vert }_1$ , where C is a positive universal constant, tends to 1 as \(g \rightarrow \infty \) $g\to \infty$ , provided that \(\Vert L_g\Vert _1=\sum _{i=1}^n L_g^i\) $\Vert L_g{\Vert }_1={\sum }_{i=1}^n{L}_g^i$ tends to infinity at most logarithmically in g and each \(L_g^i\) $L_g^i$ stays away from zero.