Variational Convergence of Discrete Minimal Surfaces

TL;DR

This paper proves that discrete minimal surfaces obtained through simplicial area minimization converge to smooth minimal surfaces in a strong topological sense as the mesh is refined, extending variational analysis tools.

Contribution

It establishes Kuratowski convergence of discrete to smooth minimal surfaces under simplicial refinement, advancing the understanding of variational convergence in geometric analysis.

Findings

Discrete minimal surfaces converge to smooth solutions

Convergence is in a topology stronger than uniform

Results extend variational analysis tools

Abstract

Building on and extending tools from variational analysis, we prove Kuratowski convergence of sets of simplicial area minimizers to minimizers of the smooth Douglas-Plateau problem under simplicial refinement. This convergence is with respect to a topology that is stronger than uniform convergence of both positions and surface normals.