Two-dimensional Calderon problem and flat metrics

2025/01/29 by Sharafutdinov, Vladimir A.
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2501.17471

Abstract

For a compact Riemannian manifold (M,g) with boundary ∂ M, the Diri\-chl\-et-to-Neumann operator Λ_g:C^∞(∂ M)\longrightarrow C^∞(∂ M) is defined by Λ_gf=.(∂ u)/(∂ν)|_{∂ M}, where ν is the unit outer normal vector to the boundary and u is the solution to the Dirichlet problem Δ_gu=0, u|_{∂ M}=f. Let g_∂ be the Riemannian metric on ∂ M induced by g. The Calderon problem is posed as follows: To what extent is (M,g) determined by the data (∂ M,g_∂,Λ_g)? We prove the uniqueness theorem: A compact connected two-dimensional Riemannian manifold (M,g) with non-empty boundary is determined by the data (∂ M,g_∂,Λ_g) uniquely up to conformal equivalence.

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