Stability Estimates in Determination of Non-orientable Surface from Its Dirichlet-to-Neumann Map

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