An inverse problem for the Riemannian minimal surface equation

TL;DR

This paper proves that the Dirichlet-to-Neumann map for the minimal surface equation uniquely determines a two-dimensional Riemannian manifold with boundary, revealing a new inverse problem result in geometric analysis.

Contribution

It establishes a uniqueness result for the inverse problem of determining a Riemannian surface from boundary measurements related to minimal surfaces.

Findings

Dirichlet-to-Neumann map determines the surface up to isometry

Unique determination of the Riemannian manifold from boundary data

Advances understanding of inverse problems in geometric analysis

Abstract

In this paper we consider determining a minimal surface embedded in a Riemannian manifold $\Sigma\times \mathbb{R}$. We show that if $\Sigma$ is a two dimensional Riemannian manifold with boundary, then the knowledge of the associated Dirichlet-to-Neumann map for the minimal surface equation determine $\Sigma$ up to an isometry.