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Mathematics > Differential Geometry

arXiv:1804.04243 (math)
[Submitted on 11 Apr 2018 (v1), last revised 26 Jun 2018 (this version, v2)]

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

Authors:Pedro Gaspar, Marco A. M. Guaraco
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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.
Comments: 26 pages, comments welcome. v2: expanded and detailed proof of the last result, references updated
Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)
Cite as: arXiv:1804.04243 [math.DG]
  (or arXiv:1804.04243v2 [math.DG] for this version)
  https://doi.org/10.48550/arXiv.1804.04243

Submission history

From: Pedro Gaspar [view email]
[v1] Wed, 11 Apr 2018 22:10:45 UTC (21 KB)
[v2] Tue, 26 Jun 2018 19:23:11 UTC (25 KB)
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