On stability of determination of Riemann surface from its DN-map

Abstract

Suppose that $M$ is a Riemann surface with boundary $\partial M$, $\Lambda$ is its DN-map, and $\mathcal{E}:M\to {\mathbb{C}}^n$ % ${\mathfrak{J}}_M$ is a holomorphic immersion. Let $M^{\prime }$ be diffeomorphic to $M$, $\partial M=\partial M^{\prime }$; let ${\Lambda }^{\prime }$ be the DN map of $M^{\prime }$. Let us write $M^{\prime }\in {\mathbb{M}}_t$ if $\parallel {\Lambda }^{\prime }-\Lambda {\parallel }_{H^1(\partial M)\to L_2(\partial M)}⩽t$ holds. We show that, for any holomorphic immersion $\mathcal{E}:M\to {\mathbb{C}}^n$ ($n⩾1$), the relation \begin{equation*} \sup_{M'\in \mathbb{M}_{t}}\inf_{\mathscr{E}'}d_{H}(\mathscr E'(M'),\mathscr{E}(M))\underset{t\to 0}{\longrightarrow}0, \end{equation*} holds, where $d_H$ is the Haussdorf distance in ${\mathbb{C}}^n$ and the infimum is taken over all holomorphic immersions ${\mathcal{E}}^{\prime }:M^{\prime }\mapsto {\mathbb{C}}^n$.

arXiv comment20 pages, 0 figures