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

Suppose that (M , g) is a compact Riemann surface with metric g and boundary ∂ ⁡ M , and Λ is its DN-map. Let M ′ be diffeomorphic to M, let ∂ ⁡ M = ∂ ⁡ M ′ and let Λ ′ be the DN-map of (M ′ , g ′) . Put (M ′ , g ′) ∈ 필 t if ∥ Λ ′ - Λ ∥ H 1 ⁢ (∂ ⁡ M) → L 2 ⁢ (∂ ⁡ M) ⩽ t holds. We show that, for any holomorphic immersion ℰ : M → ℂ n ( n ⩾ 1 ), the relation sup M ′ ∈ 필 t ⁡ inf ℰ ′ ⁡ d H ⁢ (ℰ ′ ⁢ (M ′) , ℰ ⁢ (M)) → t → 0 0 holds, where d H is the Hausdorff distance in ℂ n and the infimum is taken over all holomorphic immersions ℰ ′ : M ′ ↦ ℂ n . As it is known, Λ determines not the surface (M , g) but its conformal class { (M , ρ ⁢ g) ⁢ ∣ ρ > ⁢ 0 , ρ | ∂ ⁡ M = 1 } , while holomorphic immersions are determined by this class. In the mean time, (M , g) is conformally equivalent to ℰ ⁢ (M) , and (M ′ , g ′) is conformally equivalent to ℰ ′ ⁢ (M ′) . Thus, the closeness of the surfaces ℰ ′ ⁢ (M ′) and ℰ ⁢ (M) in ℂ n reflects the closeness of the corresponding conformal classes for close DN-maps. [ABSTRACT FROM AUTHOR]