Two-dimensional Calderon problem and flat metrics
Vladimir A. Sharafutdinov
TL;DR
This work analyzes the two‑dimensional Calderón problem on compact connected surfaces with boundary, showing that the DN‑map Λ_g together with boundary data (∂M, ds_g) determines the manifold and its flat conformal class up to isometry when reduced to flat metrics. By connecting DN‑map information to topology via Hodge theory, converting geometric data to holomorphic function algebras, and applying the Gelfand transform, the authors reconstruct the interior complex structure from boundary data. The approach combines Belishev’s Banach‑algebra framework with flat‑metric/conformal mapping techniques to prove that, in 2D, the data uniquely determine the surface and metric up to the natural conformal equivalence, with the flat‑metric case yielding an isometric recovery. The results clarify the role of conformal invariance in 2D and provide a rigorous pathway for recovering geometric structure from boundary impedance data, with potential extensions to related inverse problems on surfaces.
Abstract
For a compact Riemannian manifold $(M,g)$ with boundary $\partial M$, the Diri\-chl\-et-to-Neumann operator $Λ_g:C^\infty(\partial M)\longrightarrow C^\infty(\partial M)$ is defined by $Λ_gf=\left.\frac{\partial u}{\partialν}\right|_{\partial M}$, where $ν$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $Δ_gu=0,\ u|_{\partial M}=f$. Let $g_\partial$ be the Riemannian metric on $\partial M$ induced by $g$. The Calderon problem is posed as follows: To what extent is $(M,g)$ determined by the data $(\partial M,g_\partial,Λ_g)$? We prove the uniqueness theorem: A compact connected two-dimensional Riemannian manifold $(M,g)$ with non-empty boundary is determined by the data $(\partial M,g_\partial,Λ_g)$ uniquely up to conformal equivalence.