Solutions of the Allen-Cahn equation on closed manifolds in the presence of symmetry

TL;DR

This paper constructs solutions to the Allen-Cahn equation on closed manifolds whose nodal sets approximate minimal hypersurfaces with symmetric Jacobi fields, extending prior results to more general symmetric settings.

Contribution

It establishes the existence of Allen-Cahn solutions with prescribed nodal sets near symmetric minimal hypersurfaces on closed manifolds.

Findings

Solutions exist for small epsilon with nodal sets converging to the minimal hypersurface.

Extension of previous results to cases with symmetry and non-zero mean curvature.

Provides a method to link minimal hypersurfaces and Allen-Cahn solutions in symmetric settings.

Abstract

We prove that given a minimal hypersurface $\Gamma$ in a compact Riemannian manifold $M$ without boundary, if all the Jacobi fields of $\Gamma$ are generated by ambient isometries, then we can find solutions of the Allen-Cahn equation $-\varepsilon^2\Delta u +W'(u)=0$ on $M$, for sufficiently small $\varepsilon>0$, whose nodal sets converge to $\Gamma$. This extends the results of Pacard-Ritor\'e (in the case of closed manifolds and zero mean curvature).