On non-homeomorphic surfaces with close DN maps

TL;DR

The paper demonstrates that for a genus m surface with boundary, if its Dirichlet-to-Neumann map is close to that of a lower genus surface, then the shortest geodesics on its Schottky double become arbitrarily small.

Contribution

It establishes a quantitative relationship between the closeness of DN maps and the shrinking of systoles on the Schottky double, linking spectral data to geometric degeneration.

Findings

Small difference in DN maps implies small systoles on the Schottky double.

Closeness in operator norm of DN maps leads to geometric degeneration.

Results connect spectral boundary data with hyperbolic geometric properties.

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 $\Gamma$: $$\|\Lambda-\Lambda_*\|_{B(H^{1/2}(\Gamma);H^{-1/2}(\Gamma))}\to 0\,\Longrightarrow \ Sys(2M)\to 0.$$