Two-dimensional Calderón problem and flat metrics

For a compact Riemannian manifold (M, g) with boundary \(\partial M\), the Dirichlet-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 Calderón 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.