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

2023/06/24 by Dmitrii Korikov, Korikov, Dmitrii · 1 citation
Mathematics · #30F15 #35R30 #46J15 #46J20 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2306.14024

openalex publication_date 2023/06/24 · openalex created_date 2023/06/28 · openalex updated_date 2026/07/28

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↦ [(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.

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