Constructing solutions to the Bj\"orling problem for isothermic surfaces by structure preserving discretization

TL;DR

This paper develops a method to solve the Björling problem for isothermic surfaces by constructing discrete models and analyzing their limits, extending classical surface theory through structure-preserving discretization.

Contribution

It introduces a novel discrete approach to the Björling problem for isothermic surfaces, connecting discrete and continuous solutions via geometric discretization techniques.

Findings

Solutions can be obtained from discrete isothermic surfaces sampled along a curve.

The discrete-to-continuous limit is rigorously established for key transformations.

The approach generalizes classical surface theory to a discrete setting.

Abstract

In this article, we study an analog of the Bj\"orling problem for isothermic surfaces (that are more general than minimal surfaces): given a real analytic curve $\gamma$ in ${\mathbb R}^3$, and two analytic non-vanishing orthogonal vector fields $v$ and $w$ along $\gamma$, find an isothermic surface that is tangent to $\gamma$ and that has $v$ and $w$ as principal directions of curvature. We prove that solutions to that problem can be obtained by constructing a family of discrete isothermic surfaces (in the sense of Bobenko and Pinkall) from data that is sampled along $\gamma$, and 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.