The Allen-Cahn equation on closed manifolds

TL;DR

This paper investigates the variational properties of solutions to the Allen-Cahn equation on closed manifolds, revealing how solutions behave as the parameter approaches zero, and connecting solutions to minimal hypersurfaces.

Contribution

The paper extends the analogy between phase transitions and minimal hypersurfaces, showing the existence, multiplicity, and energy behavior of solutions on closed manifolds, and linking these to minimal hypersurface theory.

Findings

Number of min-max solutions tends to infinity as epsilon approaches zero.

Solutions at low energy are either stable or min-max with index 1.

Energy of solutions accumulates around minimal hypersurfaces with sublinear growth.

Abstract

We study global variational properties of the space of solutions to $-\varepsilon^2\Delta u + W'(u)=0$ on any closed Riemannian manifold $M$. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min-max and have index 1. We show that if $\varepsilon$ is not small enough, in terms of the Cheeger constant of $M$, then there are no interesting solutions. However, we show that the number of min-max solutions to the equation above goes to infinity as $\varepsilon\to 0$ and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed $\varepsilon$, as shown recently by G. Smith. We also show that the energy of the min-max solutions accumulate, as $\varepsilon \to 0$, around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with ${\rm Ric}_M>0$, the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet-Rosenberg. Finally, we prove that the min-max energy values are bounded from below by the widths of the area functional as defined by Marques-Neves.