Classification of ancient ovals in higher dimensional mean curvature flow

TL;DR

This paper classifies a specific family of ancient solutions called k-ovals in higher-dimensional mean curvature flow, showing they are essentially unique up to symmetries, and provides an alternative proof to recent classifications.

Contribution

It offers a new spectral parametrization approach to classify k-ovals, complementing recent breakthroughs and clarifying the moduli space structure of these solutions.

Findings

k-ovals are classified up to symmetries as the known family by Haslhofer and colleagues.

The moduli space of k-ovals is an open (k-1)-simplex modulo symmetry.

An alternative spectral method is developed for classification in arbitrary dimensions.

Abstract

We study compact non-selfsimilar ancient noncollapsed solutions to the mean curvature flow in $\mathbb{R}^{n+1}$, called ancient ovals. Our main result is the classification of $k$-ovals: any $k$-oval (characterized by having cylindrical blow down $\mathbb{R}^k\times S^{n-k}$ and the quadratic bending asymptotics) belongs, up to space-time rigid motions and parabolic dilations, to the family of ancient ovals constructed by Haslhofer and the second author. Assuming the nonexistence of exotic ovals (recently proved by Bamler-Lai), this yields a classification of all ancient ovals and identifies the moduli space, modulo symmetries, with an open $(k-1)$-simplex modulo the symmetry of simplex. Although these conclusions are contained in the recent breakthrough of Bamler-Lai classifying all ancient asymptotically cylindrical flows and resolving the mean convex neighborhood conjecture, we give an alternative argument for the independently obtained classification of $k$-ovals in arbitrary dimensions based on a different spectral parametrization.