Stability of determination of Riemann surface from its DN-map in terms of Teichmüller distance

Abstract

As is known, the Dirichlet-to-Neumann operator $\Lambda$ of a Riemannian surface $(M,g)$ determines the surface up to conformal equivalence class $[(M,g)]$. Such classes constitute the Teichmüller space with the distance ${\mathrm{d}}_T$. We show that the determination is continuous: $\Vert \Lambda -{\Lambda }^{\prime }{\Vert }_{H^1(\partial M)\to L_2(\partial M)}\to 0$ implies ${\mathrm{d}}_T([(M,g)],[(M^{\prime },g^{\prime })])\to 0$.

arXiv comment25 pages, 0 figures