Stability of determination of Riemann surface from its DN-map in terms of Teichm\"uller distance

TL;DR

This paper proves that the Dirichlet-to-Neumann map's stability in determining a Riemann surface's conformal class is quantitatively linked to the Teichmüller distance, ensuring small changes in the map imply small geometric changes.

Contribution

It establishes a continuity result connecting the Dirichlet-to-Neumann operator's norm difference to the Teichmüller distance between Riemann surfaces.

Findings

Small perturbations in the DN-map imply small Teichmüller distance changes.

The determination of a Riemann surface from its DN-map is stable.

Provides a quantitative link between boundary data and geometric structure.

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 Teichm\"uller space with the distance ${\rm d}_T$. We show that the determination is continuous: $\|\Lambda-\Lambda'\|_{H^1(\partial M)\to L_2(\partial M)}\to 0$ implies ${\rm d}_T([(M,g)],[(M',g')])\to 0$.