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

Abstract

Let $(M,g)$ and $(M^{\prime },g^{\prime })$ be non-orientable Riemannian surfaces with fixed boundary $\Gamma$ and fixed Euler characterictic $m$, and $\Lambda$ and ${\Lambda }^{\prime }$ be their Dirichlet-to-Neumann maps, respectively. We prove that the closeness of ${\Lambda }^{\prime }$ to $\Lambda$ in the operator norm implies the existence of of the near-conformal diffeomorphism $\beta$ between $(M,g)$ and $(M^{\prime },g^{\prime })$ which does not move the points of $\Gamma$. Hence we establish the continuity of the determination $\Lambda \mapsto [(M,g)]$, 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 $d_T([(M,g)],[(M^{\prime },g^{\prime })])$ via the operator norm of the difference ${\Lambda }^{\prime }-\Lambda$. 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.

arXiv comment45 pages, 0 figures