Local uniformization and free boundary regularity of minimal singular surfaces

TL;DR

This paper studies harmonic mappings of 2D complexes to understand minimal surfaces with singularities, proving real analyticity of free boundaries and constructing local uniformizations of singular minimal surfaces.

Contribution

It establishes the real analyticity of free boundary curves and provides a method for local uniformization of singular minimal surfaces using harmonic maps.

Findings

Proved real analyticity of free boundary curves.

Constructed local uniformizations of singular minimal surfaces.

Linked uniformization techniques to minimal surface extension theories.

Abstract

In continuing the study of harmonic mapping from 2-dimensional Riemannian simplicial complexes in order to construct minimal surfaces with singularity, we obtain an a-priori regularity result concerning the real analyticity of the free boundary curve. The free boundary is the singular set along which three disk-type minimal surfaces meet. Here the configuration of the singular minimal surface is obtained by a minimization of a weighted energy functional, in the spirit of J.Douglas' approach to the Plateau Problem. Using the free boundary regularity of the harmonic map, we construct a local uniformization of the singular surface as a parameterization of a neighborhood of a point on the free boundary by the singular tangent cone. In addition, applications of the local uniformization are discussed in relation to H.Lewy's real analytic extension of minimal surfaces.