Stability of determination of Riemann surface from its DN-map in terms of holomorphic immersions

Abstract

Abstract Suppose that ( M , g ) {(M,g)} is a compact Riemann surface with metric g and boundary ∂ ⁡ M {\partial M} , and Λ is its DN-map. Let M ′ {M^{\prime}} be diffeomorphic to M, let ∂ ⁡ M = ∂ ⁡ M ′ {\partial M=\partial M^{\prime}} and let Λ ′ {\Lambda^{\prime}} be the DN-map of ( M ′ , g ′ ) {(M^{\prime},g^{\prime})} . Put ( M ′ , g ′ ) ∈ 𝕄 t {(M^{\prime},g^{\prime})\in\mathbb{M}_{t}} if ∥ Λ ′ - Λ ∥ H 1 ⁢ ( ∂ ⁡ M ) → L 2 ⁢ ( ∂ ⁡ M ) ⩽ t {\lVert\Lambda^{\prime}-\Lambda\rVert_{H^{1}(\partial M)\to L^{2}(\partial M)}% \leqslant t} holds. We show that, for any holomorphic immersion ℰ : M → ℂ n {\mathscr{E}:M\to\mathbb{C}^{n}} ( n ⩾ 1 {n\geqslant 1} ), the relation sup M ′ ∈ 𝕄 t ⁡ inf ℰ ′ ⁡ d H ⁢ ( ℰ ′ ⁢ ( M ′ ) , ℰ ⁢ ( M ) ) → t → 0 0 \sup_{M^{\prime}\in\mathbb{M}_{t}}\inf_{\mathscr{E}^{\prime}}d_{H}(\mathscr{E}% ^{\prime}(M^{\prime}),\mathscr{E}(M))\xrightarrow{t\to 0}0 holds, where d H {d_{H}} is the Hausdorff distance in ℂ n {\mathbb{C}^{n}} and the infimum is taken over all holomorphic immersions ℰ ′ : M ′ ↦ ℂ n {\mathscr{E}^{\prime}:M^{\prime}\mapsto\mathbb{C}^{n}} . As it is known, Λ determines not the surface ( M , g ) {(M,g)} but its conformal class { ( M , ρ ⁢ g ) ⁢ ∣ ρ > ⁢ 0 , ρ | ∂ ⁡ M = 1 } , \bigl{\{}(M,\rho g)\mid\rho>0,\,\rho|_{\partial M}=1\bigr{\}}, while holomorphic immersions are determined by this class. In the mean time, ( M , g ) {(M,g)} is conformally equivalent to ℰ ⁢ ( M ) {\mathscr{E}(M)} , and ( M ′ , g ′ ) {(M^{\prime},g^{\prime})} is conformally equivalent to ℰ ′ ⁢ ( M ′ ) {\mathscr{E}^{\prime}(M^{\prime})} . Thus, the closeness of the surfaces ℰ ′ ⁢ ( M ′ ) {\mathscr{E}^{\prime}(M^{\prime})} and ℰ ⁢ ( M ) {\mathscr{E}(M)} in ℂ n {\mathbb{C}^{n}} reflects the closeness of the corresponding conformal classes for close DN-maps.