Existence of monotone Morse flow lines of the expander functional

TL;DR

This paper constructs a monotone Morse flow line connecting an unstable asymptotically conical self-expander to a stable one, demonstrating the existence of such flows with controlled singularities.

Contribution

It introduces a method to construct singular Morse flow lines of the expander functional linking unstable and stable self-expanders.

Findings

Existence of monotone Morse flow lines between self-expanders.

Flow lines have small singular sets.

Flow connects unstable to stable self-expanders.

Abstract

Given a smooth asymptotically conical self-expander that is strictly unstable we construct a (singular) Morse flow line of the expander functional that connects it to a stable self-expander. This flow is monotone in a suitable sense and has small singular set.