On the Existence and Uniqueness of Ancient Rescaled Mean Curvature Flows

TL;DR

This paper establishes the existence of ancient solutions to rescaled mean curvature flow originating from asymptotically conical self-expanders, and classifies flows from generic cones with low entropy in low dimensions.

Contribution

It proves existence and uniqueness of ancient solutions from asymptotically conical self-expanders and classifies flows from generic low-entropy cones.

Findings

Existence of ancient solutions from asymptotically conical self-expanders.

Strong uniqueness theorem for generic cones.

Classification of flows from low-entropy cones in low dimensions.

Abstract

We show existence of ancient solutions to the rescaled mean curvature flow starting from a given asymptotically conical self-expander. These are examples of mean curvature flows coming out of cones that are not self-similar. We also show a strong uniqueness theorem when the cone is generic and use it to classify mean curvature flows coming out of generic cones of small entropy in low dimensions.