On the Topology of Surfaces with a Common Boundary and Close DN-Maps

Let Ω be a smooth compact Riemann surface with the boundary Γ, and let Λ : H¹(Γ) ↦ L₂(Γ), Λf ≔ ∂_νu|_Γ be its DN map, where u obeys Δ_gu = 0 in Ω and u = f on Γ. As is known, the genus m of the surface Ω is determined by its DN map Λ. In this paper, we prove the existence of Riemann surfaces of arbitrary genus m′ > m, with the boundary Γ, and such that their DN maps are arbitrarily close to Λ with respect to the operator norm. In other words, an arbitrarily small perturbation of the DN map may change the surface topology.