Self-expanders of positive genus

TL;DR

This paper constructs new self-expanding surfaces of positive genus in three-dimensional space, demonstrating their use in mean curvature flow with decreasing genus and analyzing their asymptotic behavior.

Contribution

It introduces a method to construct self-expanders of positive genus asymptotic to cones, advancing understanding of mean curvature flow singularities.

Findings

Constructed self-expanders of positive genus asymptotic to cones.

Demonstrated mean curvature flow with genus decreasing but not vanishing.

Analyzed asymptotic behavior of a sequence of self-expanders with unbounded genus.

Abstract

For a general class of cones in $\mathbb{R}^3$, we construct self-expanders of positive genus asymptotic to these cones. As a result, we use these self-expanders to construct a mean curvature flow with genus strictly decreasing but not to zero at the first singular time. We also construct a sequence of self-expanders with unbounded genus which are asymptotic to the same rotationally symmetric cone. Moreover, we characterize the asymptotic behavior of the sequence.