Two-dimensional Calderon problem and flat metrics

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