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Solution of the Björling problem by discrete approximation

Ulrike Bücking, Daniel Matthes

TL;DR

The authors address the local Björling problem for minimal surfaces by constructing discrete minimal surfaces via CR-maps that preserve cross-ratios on a rectangular lattice. They translate Björling data into discrete Weierstrass data through a holomorphic reparametrization φ and a holomorphic function g, then evolve a discrete cross-ratio equation to obtain G_{m,n} approximating g with $C^∞$ accuracy and $O(ε^2)$ error for the resulting surfaces. The core contributions are (i) a concrete scheme to generate discrete CR-mappings from Björling data, (ii) a rigorous convergence analysis showing $F^ε o f$, α, β → g', and hence ${ rak F}_{m,n} o { rak F}$ in $C^ty$, and (iii) two practical initial-data strategies enabling local construction of the discrete minimal surfaces. The results provide a provably accurate, local discretization framework for minimal surfaces solving Björling-type problems, with potential applications in geometric modeling and numerical differential geometry.

Abstract

The Björling problem amounts to the construction of a minimal surface from a real-analytic curve with a given real-analytic normal vector field. We approximate that solution locally by discrete minimal surfaces as special discrete isothermic surfaces (as defined by Bobenko and Pinkall in 1996). The main step in our construction is the approximation of the sought surface's Weierstrass data by discrete conformal maps. We prove that the approximation error is of the order of the square of the mesh size.

Solution of the Björling problem by discrete approximation

TL;DR

The authors address the local Björling problem for minimal surfaces by constructing discrete minimal surfaces via CR-maps that preserve cross-ratios on a rectangular lattice. They translate Björling data into discrete Weierstrass data through a holomorphic reparametrization φ and a holomorphic function g, then evolve a discrete cross-ratio equation to obtain G_{m,n} approximating g with accuracy and error for the resulting surfaces. The core contributions are (i) a concrete scheme to generate discrete CR-mappings from Björling data, (ii) a rigorous convergence analysis showing , α, β → g', and hence in , and (iii) two practical initial-data strategies enabling local construction of the discrete minimal surfaces. The results provide a provably accurate, local discretization framework for minimal surfaces solving Björling-type problems, with potential applications in geometric modeling and numerical differential geometry.

Abstract

The Björling problem amounts to the construction of a minimal surface from a real-analytic curve with a given real-analytic normal vector field. We approximate that solution locally by discrete minimal surfaces as special discrete isothermic surfaces (as defined by Bobenko and Pinkall in 1996). The main step in our construction is the approximation of the sought surface's Weierstrass data by discrete conformal maps. We prove that the approximation error is of the order of the square of the mesh size.
Paper Structure (21 sections, 13 theorems, 118 equations, 8 figures)

This paper contains 21 sections, 13 theorems, 118 equations, 8 figures.

Table of Contents

  1. Introduction
  2. From Björling data to Cauchy data for Weierstrass representation
  3. Representation formulas
  4. Determination of $\phi$ and $g$ from ${\mathfrak F}_0$ and ${\mathfrak N}_0$
  5. Construction of rectangular lattices and discrete holomorphic functions
  6. Construction of Cauchy data for $G_{m,n}$ for the proof of Theorem \ref{['theo:MinimalConv']}
  7. Cross-ratio evolution: (discrete) mappings from Cauchy data
  8. Derivation of a discrete evolution equation
  9. Consistency of the discrete evolution equation \ref{['eq:devolveFmn']}
  10. $C^\infty$-convergence of $F^\varepsilon$ to $f$
  11. Some useful norms and their basis properties
  12. Existence of a continuous solution of \ref{['eq:evolvef']}
  13. Approximation in $C^0$
  14. Smooth convergence
  15. Convergence of the CR-mappings $G$ and of the discrete minimal surfaces from Björling data
  16. ...and 6 more sections

Key Result

Theorem 1

Given Björling data ${\mathfrak F}_0$ and ${\mathfrak N}_0$ and a point ${\mathfrak F}_0(t_0)$ such that $\dot{\mathfrak F}_0(t_0)$ is not parallel to $\dot{\mathfrak N}_0(t_0)$, we can locally approximate the solution of the Björling problem ${\mathfrak F}$ by discrete minimal surfaces ${\mathfrak

Figures (8)

  • Figure 1: Example of the parametrized curve $\phi|_{[-\hat{a},\hat{a}]} =:\gamma:[-\hat{a},\hat{a}]\to\Omega$ with $t\mapsto u(t)+i v(t)$ which gives rise to a coordinate transformation from the $(t,\eta)$-plane to the $(u,v)$-plane.
  • Figure 2: Example of the lattice $\Omega^\varepsilon$ (right) and the curve $\gamma=\phi|_{[-\hat{a},\hat{a}]}$ (right, colored green) passing through the lattice points $p_{n,n}$.
  • Figure 3: Example of a CR-mapping $G_{m,n}$ on a rectangular lattice
  • Figure 4: Example of an initial 'zig-zag'-curve in parameter space (black points), containing points of the given curve (green), for the evolution of CR-mappings
  • Figure 5: Left: Given curve ${\mathfrak F}_0$ (red) and black points of the discrete minimal surface ${\mathfrak F}_{m,n}$ obtained from our construction procedure in Section \ref{['secConstruction']}. Right: Image of the discrete holomorphic map $G_{m,n}$ (black points) and curve of initial values $G_0$ (red).
  • ...and 3 more figures

Theorems & Definitions (24)

  • Theorem 1
  • Lemma 1
  • Remark 1
  • Definition 1: book, see also BS08
  • Remark 2
  • Theorem 2
  • Theorem 3
  • Lemma 2
  • Remark 3
  • Lemma 3
  • ...and 14 more