Generic nondegeneracy for solutions of the Allen-Cahn equation under a volume constraint in closed manifolds

TL;DR

This paper proves that, generically, solutions to the Allen-Cahn equation with a volume constraint on closed manifolds are nondegenerate, extending previous techniques to a broader geometric setting.

Contribution

It adapts existing methods to establish generic nondegeneracy of solutions for the constrained Allen-Cahn equation on closed manifolds.

Findings

Proves generic nondegeneracy of solutions

Extends techniques to manifold setting

Addresses volume constraint in solutions

Abstract

Let $M^n$ be a connected closed smooth manifold, where $n\geq 2$. In this article, we adapt the techniques in Micheletti and Pistoia (2009) and Ghimenti and Micheletti (2011) to prove the generic nondegeneracy for solutions of the Van der Waals-Allen-Cahn-Hilliard equation under a volume constraint in $M$.