Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders

TL;DR

This paper proves a Liouville-type theorem for certain mean-convex self-expanders of the mean curvature flow and demonstrates their rotational symmetry when asymptotic to specific cones.

Contribution

It establishes a Liouville-type theorem and shows rotational symmetry of asymptotically conical mean-convex self-expanders, advancing understanding of their geometric structure.

Findings

Liouville-type theorem for mean-convex self-expanders

Rotational symmetry of asymptotic self-expanders

Characterization of self-expanders with decaying principal curvatures

Abstract

In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to $O(n)$-invariant cones are rotationally symmetric.