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

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 $\mathbb{S}^3$.