Two-dimensional Calderon problem and flat metrics

TL;DR

This paper proves that in two dimensions, a compact Riemannian manifold with boundary is uniquely determined up to conformal equivalence by its boundary metric and Dirichlet-to-Neumann operator, solving a special case of the Calderon problem.

Contribution

It establishes the uniqueness of the inverse boundary value problem for 2D Riemannian manifolds up to conformal maps, a significant step in inverse geometry.

Findings

Unique determination of 2D manifolds from boundary data

Conformal equivalence class characterization

Advances understanding of Calderon problem in 2D

Abstract

For a compact Riemannian manifold $(M,g)$ with boundary $\partial M$, the Diri\-chl\-et-to-Neumann operator $\Lambda_g:C^\infty(\partial M)\longrightarrow C^\infty(\partial M)$ is defined by $\Lambda_gf=\left.\frac{\partial u}{\partial\nu}\right|_{\partial M}$, where $\nu$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $\Delta_gu=0,\ u|_{\partial M}=f$. Let $g_\partial$ be the Riemannian metric on $\partial M$ induced by $g$. The Calderon problem is posed as follows: To what extent is $(M,g)$ determined by the data $(\partial M,g_\partial,\Lambda_g)$? We prove the uniqueness theorem: A compact connected two-dimensional Riemannian manifold $(M,g)$ with non-empty boundary is determined by the data $(\partial M,g_\partial,\Lambda_g)$ uniquely up to conformal equivalence.