Expanders for Mean Curvature Flow and Counterexamples to Ilmanen's Genus-Reduction Conjecture

TL;DR

This paper constructs new mean curvature flow expanders asymptotic to specific cones, demonstrating that the genus of the resulting expanders can be arbitrarily larger than that of the initial shrinker, countering Ilmanen's conjecture.

Contribution

It introduces new expanders for mean curvature flow with arbitrary genus, providing counterexamples to the genus-reduction conjecture.

Findings

Existence of expanders asymptotic to certain cones.

Ability to produce expanders with arbitrarily large genus.

Counterexamples to Ilmanen's genus-reduction conjecture.

Abstract

We construct new expanders for mean curvature flow that are smoothly asymptotic to cones arising from certain shrinkers. For each such cone, we prove the existence of expanders of arbitrarily large genus. Thus, for a fixed incoming shrinker, the genus of the outgoing expander can be chosen much larger than the genus before the singularity, contrary to Ilmanen's genus-reduction conjecture.