Low complexity solutions of the Allen-Cahn equation on three-spheres

TL;DR

This paper proves the existence of at least four low-index solutions to the Allen-Cahn equation with spherical interfaces on three-spheres with bumpy metrics, using recent mathematical results.

Contribution

It establishes the existence of multiple solutions with controlled index for the Allen-Cahn equation on three-spheres, extending previous theoretical understanding.

Findings

At least four solutions exist with index at most two.

Solutions have spherical interfaces.

Results apply to three-spheres with any bumpy metric.

Abstract

In this short note, we prove that on the three-sphere with any bumpy metric there exist at least four solutions of the Allen-Cahn equation with spherical interface and index at most two. The proof combines several recent results from the literature.