Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere

TL;DR

This paper explores the connection between mean curvature flow and low energy solutions of the Allen-Cahn equation on the 3-sphere, constructing eternal flows linking Clifford tori and equatorial spheres.

Contribution

It constructs and analyzes eternal solutions to the Allen-Cahn equation on the 3-sphere, linking geometric flows with phase transition models and utilizing recent classification results.

Findings

Constructed eternal integral Brakke flows connecting Clifford tori to spheres.

Analyzed symmetry properties of these flows.

Linked Allen-Cahn solutions to mean curvature flow via singular limits.

Abstract

In this article we study eternal solutions to the Allen-Cahn equation in the 3-sphere, in view of the connection between the gradient flow of the associated energy functional, and the mean curvature flow. We construct eternal integral Brakke flows that connect Clifford tori to equatorial spheres, and study a family of such flows, in particular their symmetry properties. Our approach is based on the realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, as studied by Ilmanen, and Tonegawa, and it uses the classification of ancient gradient flows in spheres, by K. Choi and C. Mantoulidis, as well as the rigidity of stationary solutions with low Morse index proved by F. Hiesmayr.