On the construction of closed nonconvex nonsoliton ancient mean curvature flows

TL;DR

This paper constructs new examples of closed, embedded, ancient mean curvature flows with toroidal topology in all dimensions, which are neither mean convex nor solitons, expanding the understanding of such flows beyond known self-shrinkers.

Contribution

It introduces the first known closed, embedded, non-soliton ancient mean curvature flows with toroidal topology in any dimension, using perturbation analysis of existing self-shrinking solutions.

Findings

Constructed ancient flows with $S^1 \times S^{n-1}$ topology for all $n \ge 2

Examples are not mean convex and not solitons

Analysis based on perturbations of known self-shrinking doughnuts

Abstract

We construct closed, embedded, ancient mean curvature flows in each dimension $n\ge 2$ with the topology of $S^1 \times S^{n-1}$. These examples are not mean convex and not solitons. They are constructed by analyzing perturbations of the self-shrinking doughnuts constructed by Drugan and Nguyen (or, alternatively, Angenent's self shrinking torus when $n =2$)