The Calder\'on problem on Riemannian surfaces and of minimal surfaces

TL;DR

This paper proves that the Dirichlet-Neumann map on a Riemannian surface uniquely determines its geometric structure and applies this to identify minimal surfaces in 3D manifolds, advancing inverse geometric problems.

Contribution

It introduces new techniques to determine the boundary trace of holomorphic functions from the Dirichlet-Neumann map, removing previous assumptions of planarity or known geometry.

Findings

Dirichlet-Neumann map determines surface topology, geometry, and metric.

Volumes of embedded minimal surfaces determine their structure.

New Carleman estimate techniques for inverse problems.

Abstract

In this paper we prove two results. The first shows that the Dirichlet-Neumann map of the operator $\Delta_g+q$ on a Riemannian surface can determine its topological, differential, and metric structure. Earlier work of this type assumes a priori that the surface is a planar domain [36] or that the geometry is a priori known [29]. We will then apply this result to study a geometric inverse problem for determining minimal surfaces embedded in $3$-dimensional Riemannian manifolds. In particular we will show that knowledge of the volumes of embedded minimal surfaces determine not only their topological and differential structure but also their Riemannian structure as an embedded hypersurface. Such geometric inverse problems are partially inspired by the physical models proposed by the AdS/CFT correspondence. The crucial ingredient in removing the planar domain assumption is the determination of the boundary trace of holomorphic functions from knowledge of the Dirichlet-Neumann map of $\Delta_g +q$. This requires a new type of argument involving Carleman estimates and construction of CGO whose phase functions are not Morse as in the case of [29]. We anticipate that these techniques could be of use for studying other inverse problems in geometry and PDE.