Unique asymptotics of ancient convex mean curvature flow solutions

TL;DR

This paper characterizes the unique asymptotic behavior of ancient convex solutions to mean curvature flow with symmetry, providing precise descriptions and confirming predictions for specific solutions.

Contribution

It establishes the uniqueness of asymptotics for a class of ancient convex solutions with symmetry and offers detailed asymptotic descriptions, confirming prior predictions.

Findings

All such solutions have unique asymptotics as time approaches negative infinity.

The paper provides a precise asymptotic description of these solutions.

Confirmed asymptotics for solutions constructed by White, and Haslhofer and Hershkovits.

Abstract

We study the compact noncollapsed ancient convex solutions to Mean Curvature Flow in $\mathbb{R}^{n+1}$ with $O(1)\times O(n)$ symmetry. We show they all have unique asymptotics as $t\to -\infty$ and we give precise asymptotic description of these solutions. In particular, solutions constructed by White, and Haslhofer and Hershkovits have those asymptotics (in the case of those particular solutions the asymptotics was predicted and formally computed by Angenent).