Stability estimates in determination of non-orientable surface from its Dirichlet-to-Neumann map

Abstract

Let $(M,g)$ and $(M',g')$ be non-orientable Riemannian surfaces with fixed boundary $Γ$ and fixed Euler characterictic $m$, and $Λ$ and $Λ'$ be their Dirichlet-to-Neumann maps, respectively. We prove that the closeness of $Λ'$ to $Λ$ in the operator norm implies the existence of of the near-conformal diffeomorphism $β$ between $(M,g)$ and $(M',g')$ which does not move the points of $Γ$. Hence we establish the continuity of the determination $Λ\mapsto [(M,g)]$, where $[(M,g)]$ is the conformal class of $(M,g)$ and the set of such conformal classes is endowed with the natural Teichmüller-type metric $d_T$. In both orientable and non-orientable case we provide quantitative estimates of $d_T([(M,g)],[(M',g')])$ via the operator norm of the difference $Λ'-Λ$. We also obtain generalizations of the results above to the case in which the Dirichlet-to-Neumann map is given only on a segment of the boundary.