Noncompact self-shrinkers for mean curvature flow with arbitrary genus

TL;DR

This paper proves the existence of noncompact self-shrinkers with arbitrary genus using min-max methods, analyzes their asymptotic behavior, and provides numerical evidence for additional families with specific geometric features.

Contribution

It offers a rigorous existence proof for noncompact self-shrinkers of arbitrary genus, confirming conjectures and analyzing their asymptotic structure.

Findings

Existence of noncompact self-shrinkers with arbitrary genus.

Confirmation that large genus self-shrinkers have one asymptotically conical end.

Numerical evidence for self-shrinkers with odd genus and two ends.

Abstract

In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.