Self-shrinkers with any number of ends in $\mathbb{R}^{3}$ by stacking $\mathbb{R}^{2}$

TL;DR

This paper constructs new self-shrinker surfaces in three-dimensional space with multiple ends and complex topology by stacking planes connected with catenoidal bridges, using PDE gluing techniques inspired by minimal surface theory.

Contribution

It introduces a novel PDE gluing method to explicitly construct self-shrinkers with arbitrary numbers of ends and genus, extending previous minimal surface doubling techniques.

Findings

Constructed self-shrinkers with 2J+1 ends and genus 2J(m-1)

Surfaces resemble stacked planes connected by catenoidal bridges

Method based on Linearised Doubling approach from minimal surface theory

Abstract

For each half-integer $J$ and large enough integer $m$ we construct by PDE gluing methods a self-shrinker $\breve{M}[J,m]$ with $2J+1$ ends and genus $2J(m-1)$. $\breve{M}[J,m]$ resembles the stacking of $2J+1$ levels of the plane $\mathbb{R}^2$ in $\mathbb{R}^3$ that have been connected by $2Jm$ catenoidal bridges with $m$ bridges connecting each pair of adjacent levels. It observes the symmetry of an $m$-gonal prism (when $J$ is a half integer) or an $m$-gonal antiprism (when $J$ is an integer). The construction is based on the Linearised Doubling (LD) methodology which was first introduced by Kapouleas in the construction of minimal surface doublings of $\mathbb{S}^2_{eq}$ in $\mathbb{S}^3$.