Swallowtails and cone-like singularities on maxface

TL;DR

This paper constructs a sequence of maxfaces with increasing swallowtails that converges to a given maxface with cone-like singularities, advancing understanding of singularity structures in maxfaces.

Contribution

It introduces a method to approximate maxfaces with cone-like singularities using sequences with more swallowtails, revealing new insights into their geometric structure.

Findings

Sequence of maxfaces with more swallowtails converges to the original maxface.

Provides a framework for understanding singularity transitions in maxfaces.

Enhances the classification of singularities in maxface geometry.

Abstract

When a connected component of the set of singular points of the maxface $X$ consists of only generalized cone-like singular points, we construct a sequence of maxfaces $X_n$, with an increasing number of swallowtails, converging to the maxface $X$. We include the general discussion toward this.