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

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.