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Main Title: Constructing solutions to the Björling problem for isothermic surfaces by structure preserving discretization
Author(s): Bücking, Ulrike
Matthes, Daniel
Type: Book Part
Language Code: en
Is Part Of: 10.1007/978-3-662-50447-5
Abstract: In this article, we study an analog of the Björling problem for isothermic surfaces (that are a generalization of minimal surfaces): given a regular curve γ in R3 and a unit normal vector field n along γ, find an isothermic surface that contains γ, is normal to n there, and is such that the tangent vector γ′ bisects the principal directions of curvature. First, we prove that this problem is uniquely solvable locally around each point of γ, provided that γ and n are real analytic. The main result is that the solution can be obtained by constructing a family of discrete isothermic surfaces (in the sense of Bobenko and Pinkall) from data that is read off from γ, and then passing to the limit of vanishing mesh size. The proof relies on a rephrasing of the Gauss-Codazzi-system as analytic Cauchy problem and an in-depth-analysis of its discretization which is induced from the geometry of discrete isothermic surfaces. The discrete-to-continuous limit is carried out for the Christoffel and the Darboux transformations as well.
Issue Date: 2016
Date Available: 1-Sep-2017
DDC Class: 510 Mathematik
Book Title: Advances in discrete differential geometry
Editor: Bobenko, Alexander I.
Publisher: Springer
Publisher Place: Berlin, Heidelberg
Publisher DOI: 10.1007/978-3-662-50447-5_10
Page Start: 309
Page End: 345
ISBN: 978-3-662-50447-5
Appears in Collections:Inst. Mathematik » Publications

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