On complete embedded translating solitons of the mean curvature flow that are of finite genus

TL;DR

This paper constructs new complete embedded translating solitons for the mean curvature flow by desingularizing specific geometric unions, resulting in surfaces with three ends and arbitrary finite genus.

Contribution

It introduces a novel desingularization method to produce complete embedded translating solitons with prescribed topological and geometric properties.

Findings

Successfully desingularized unions of Grim paraboloids and Costa-Hoffman-Meeks surfaces.

Produced new examples of translating solitons with three ends and arbitrary finite genus.

Expanded the understanding of the topology and geometry of mean curvature flow solitons.

Abstract

We desingularise the union of $3$ Grim paraboloids along Costa-Hoffman-Meeks surfaces in order to obtain complete embedded translating solitons of the mean curvature flow with $3$ ends and arbitrary finite genus.