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

TL;DR

This paper proves that small changes in the Dirichlet-to-Neumann map of a Riemann surface lead to small Hausdorff distance variations in the images of holomorphic immersions, establishing stability in the inverse boundary value problem.

Contribution

It demonstrates a stability result for the inverse problem of determining a Riemann surface from its DN-map using holomorphic immersions.

Findings

Small perturbations in DN-map imply small Hausdorff distance changes in surface images.

Stability holds uniformly for all holomorphic immersions.

The result applies to surfaces with boundary and their holomorphic embeddings.

Abstract

Suppose that $M$ is a Riemann surface with boundary $\partial M$, $\Lambda$ is its DN-map, and $\mathscr E:M\to\mathbb{C}^{n}$ % $\mathfrak{J}_{M}$ is a holomorphic immersion. Let $M'$ be diffeomorphic to $M$, $\partial M=\partial M'$; let $\Lambda'$ be the DN map of $M'$. Let us write $M'\in\mathbb M_t$ if $\parallel\Lambda'-\Lambda\parallel_{H^{1}(\partial M)\to L_{2}(\partial M)}\leqslant t$ holds. We show that, for any holomorphic immersion $\mathscr{E}: M \to \mathbb C^n$ ($n\geqslant 1$), 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 $\mathscr E': M'\mapsto\mathbb C^n$.