Approximating singularities by a cuspidal edge on a maxface

TL;DR

This paper establishes conditions for singularities in maxfaces and demonstrates the existence of maxfaces with cuspidal edges near certain singularities, including examples with diverse singularity types.

Contribution

It provides necessary and sufficient conditions for singularities in maxfaces and constructs examples with cuspidal edges and various singularity types.

Findings

Conditions for singularities in maxfaces are characterized.

Existence of maxfaces with cuspidal edges near certain singularities.

Examples illustrating diverse singularity types in maxfaces.

Abstract

We give necessary and sufficient conditions on the singular Bj\"{o}rling data to the singular Bj\"{o}rling problem's solution has a prescribed nature of singularity. As an application, we prove that near a maxface with a particular type of singularity, there is another maxface having a cuspidal-edge. An example of a maxface having various types of singularities is also given.