Two-dimensional Calderon problem and flat metrics

Abstract

For a compact Riemannian manifold $(M,g)$ with boundary $\partial M$, the Diri\-chl\-et-to-Neumann operator $\hat{I}{›}_g:C^{\infty }(\partial M)⟶C^{\infty }(\partial M)$ is defined by $\hat{I}{›}_{g}f={\frac{\partial u}{\partial \hat{I}½}|}_{\partial M}$, where $\hat{I}½$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $\hat{I}{”}_{g}u=0,u{|}_{\partial M}=f$. 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 $(\partial M,g_{\partial },\hat{I}{›}_g)$? We prove the uniqueness theorem: A compact connected two-dimensional Riemannian manifold $(M,g)$ with non-empty boundary is determined by the data $(\partial M,g_{\partial },\hat{I}{›}_g)$ uniquely up to conformal equivalence.