Ancient solutions of the mean curvature flow

TL;DR

This paper constructs ancient solutions to the mean curvature flow that evolve from oval shapes at negative infinity to round points at t=0, and characterizes the round sphere among such flows using recent estimates.

Contribution

It proves the existence of specific ancient solutions with detailed asymptotic behaviors and characterizes the round shrinking sphere among ancient flows.

Findings

Existence of ancient solutions with oval asymptotics

Asymptotic models near the center and tips of solutions

Characterization of round shrinking sphere among ancient flows

Abstract

In this short article, we prove the existence of ancient solutions of the mean curvature flow that for t -> 0 collapse to a round point, but for t -> -infinity become more and more oval: near the center they have asymptotic shrinkers modeled on round cylinders S^j x R^n-j and near the tips they have asymptotic translators modeled on Bowl^j+1 x R^n-j-1. We also give a characterization of the round shrinking sphere among ancient alpha-Andrews flows. Our proofs are based on the recent estimates of Haslhofer-Kleiner.