Mean curvature flows with prescribed singular sets

TL;DR

This paper constructs mean-convex ancient solutions to mean curvature flow in high dimensions with prescribed singular sets, demonstrating control over singularity formation in geometric flows.

Contribution

It introduces a method to create mean-convex ancient solutions with any given closed set as the singular set, extending understanding of singularity formation.

Findings

Constructed solutions with prescribed singular sets in mean curvature flow.

Demonstrated solutions exist in arbitrarily close Euclidean metrics.

Extended the class of known ancient solutions with controlled singularities.

Abstract

For every closed set $K \subset \mathbb{R}^n$ and every $m \geq 2$, we construct a mean-convex ancient solution to mean curvature flow of hypersurfaces in $\mathbb{R}^{m+n}$, with respect to a smooth Riemannian metric arbitrarily $C^\infty$-close to the Euclidean metric, whose first-time singular set is exactly $K \times \{0\}$.