Determination of Riemannian surface via boundary data: algebraic approach

The Dirichlet-to-Neumann map of a Riemannian surface $(M,g)$ with the boundary $\Gamma$ is given by $\Lambda: \ f\mapsto \partial_\nu u^f|_\Gamma$, where $u^f$ is a harmonic function in $M$ with the trace $f$ on $\Gamma$ and $\nu$ is the outward normal to $\Gamma$. We discuss the algebraic approach for determining the unknown $(M,g)$ via its DN map $\Lambda$. Also, the characterization of DN-operators is provided, and a continuous (in a relevant sense) dependence of the surface $(M,g)$ on its DN map $\Lambda$ is established. The key instrument is the algebra of holomorphic functions on $(M,g)$. The approach is generalized for the cases of non-orientable surfaces and surfaces with internal holes.