TY - CHAP

T1 - Constructing solutions to the björling problem for isothermic surfaces by structure preserving discretization

AU - Bücking, Ulrike

AU - Matthes, Daniel

N1 - Publisher Copyright:
© The Editor(s) (if applicable) and The Author(s) 2016.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85006817409&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-50447-5_10

DO - 10.1007/978-3-662-50447-5_10

M3 - Chapter

AN - SCOPUS:85006817409

SN - 9783662504468

SP - 309

EP - 345

BT - Advances in Discrete Differential Geometry

PB - Springer Berlin Heidelberg

ER -