The Weyl Law for the phase transition spectrum and density of limit interfaces

TL;DR

This paper establishes a Weyl Law for the phase transition spectrum and applies it to prove density and equidistribution results for minimal hypersurfaces, providing alternative proofs of Yau's conjecture using the Allen-Cahn approach.

Contribution

It introduces a Weyl Law for the phase transition spectrum and adapts existing minimal hypersurface results to the phase transition setting, offering new proofs of Yau's conjecture.

Findings

Weyl Law for phase transition spectrum proved

Density and equidistribution of minimal hypersurfaces established

Alternative proofs of Yau's conjecture provided

Abstract

We prove a Weyl Law for the phase transition spectrum based on the techniques of Liokumovich-Marques-Neves. As an application we give phase transition adaptations of the proofs of the density and equidistribution of minimal hypersufaces for generic metrics by Irie-Marques-Neves and Marques-Neves-Song, respectively. We also prove the density of separating limit interfaces for generic metrics in dimension 3, based on the recent work of Chodosh-Mantoulidis, and for generic metrics on manifolds containing only separating minimal hypersurfaces, e.g. $H_n(M,\mathbb{Z}_2) = 0$, for $4 \leq n+1 \leq 7$. These provide alternative proofs of Yau's conjecture on the existence of infinitely many minimal hypersurfaces for generic metrics on each setting, using the Allen-Cahn approach.