Maxfaces with infinitely many Swallowtails

TL;DR

This paper constructs and analyzes infinite genus maxfaces with multiple swallowtails and planar ends, revealing new families with specific singularity and embedding properties.

Contribution

It introduces new infinite genus maxface families with prescribed singularities, swallowtail counts, and embedding characteristics, expanding the understanding of maxface geometry.

Findings

Existence of period-2 maxface family with 4 swallowtails per neck

Existence of period-3 maxface family with 24 swallowtails per period

Construction of maxfaces with infinitely many planar ends and swallowtails

Abstract

In this article, we discuss the existence of a 1-parameter infinite genus family of maxfaces having infinitely many planar (spacelike) ends and infinitely many swallowtails. In particular, we show the existence of the following: (1) a period-2 family of maxfaces with infinitely many planar ends and alternating singularity types, where every odd-layer neck has exactly four swallowtails, while each even-layer neck is almost conical; for $n=2$, the fundamental piece is a genus-1 Wei-type maxface; (2) a period-3 family where every neck carries four swallowtails (24 per period); and (3) a period-2 family of maxfaces with an almost conical singularity on every neck. All maxfaces are embedded in a wider sense.