Constructing Solutions to the Björling Problem for Isothermic Surfaces by Structure Preserving Discretization

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 $$\gamma $$ γ in $$\mathbb {R}^3$$ R 3 and a unit normal vector field n along $$\gamma $$ γ , find an isothermic surface that contains $$\gamma $$ γ , is normal to n there, and is such that the tangent vector $$\gamma '$$ γ ′ bisects the principal directions of curvature. First, we prove that this problem is uniquely solvable locally around each point of $$\gamma $$ γ , provided that $$\gamma $$ 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 $$\gamma $$ 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.