November 17, 2022. D. V. Korikov (POMI). Determination of Riemannian surface via boundary data: algebraic approach.

The Dirichlet-to-Neumann map of a Riemannian surface (M,g) with the boundary Γ is given by Λ: f ↦ ∂_νu^f|_Γ, where u^f is a harmonic function in M with the trace f on Γ and ν is the outward normal to Γ. We discuss the algebraic approach for determining the unknown (M,g) via its DN map Λ. 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 Λ 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.

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