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The Weyl Law for the phase transition spectrum and density of limit interfaces
Geometric and Functional Analysis ( IF 2.2 ) Pub Date : 2019-04-17 , DOI: 10.1007/s00039-019-00489-1
Pedro Gaspar , Marco A. M. Guaraco

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.

中文翻译:

韦尔定律的极限界面相变谱和密度

我们基于Liokumovich–Marques–Neves的技术,证明了针对相变谱的Weyl定律。作为应用,我们分别由Irie–Marques–Neves和Marques–Neves–Song给出了通用度量的最小超曲面的密度和等式证明的相变适应。我们还基于Chodosh–Mantoulidis的最新工作以及仅包含最小极小曲面分离的流形上的通用度量,证明了维度3中通用度量的极限界面分离密度,例如\({H_ {n}(M,\ mathbb {Z} _2)= 0} \),用于\({4 \ leq n + 1 \ leq 7} \)。这些方法使用Allen-Cahn方法,为每种情况下通用度量的无限多个最小超曲面的存在提供了丘(Yau)猜想的替代证明。
更新日期:2019-04-17
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