On non-homeomorphic surfaces with close DN maps

Abstract

Let $(M,g)$ be a genus $m$ surface with boundary $\Gamma$ and DN map $\Lambda$. Introduce the Schottky double $2M$ of $(M,g)$ and denote by $Sys(2M)$ the length of the shortest closed geodesics in the hyperbolic metrics on $2M$. We prove that $Sys(2M)$ is small if $\Lambda$ is close, in the operator norm, to the DN map $\Lambda_*$ of some surface $(M_*,g_*)$ of lower genus $m_*<m$ with the same boundary $\Gamma$: $$\|\Lambda-\Lambda_*\|_{B(H^{1/2}(\Gamma);H^{-1/2}(\Gamma))}\to 0\,\Longrightarrow \ Sys(2M)\to 0.$$