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

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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.

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Acknowledgements

Both authors would like to thank Fernando C. Marques and André Neves for useful discussions and their interest in this work. The first author is grateful to the Department of Mathematics at Princeton University for its hospitality. Part of this work and the first drafts were carried out while visiting during the academic year of 2017–2018. The second author would like to thank FIM - ETH, Zurich for their kind hospitality, where this work was finished during a visit in Spring 2018.

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  1. Department of Mathematics, The University of Chicago, 5734 S University Ave, Chicago, IL, 60637, USA

    Pedro Gaspar & Marco A. M. Guaraco

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  1. Pedro Gaspar
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  2. Marco A. M. Guaraco
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Correspondence to Marco A. M. Guaraco.

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The first author was partly supported by NSF Grant DMS-1311795.

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Gaspar, P., Guaraco, M.A.M. The Weyl Law for the phase transition spectrum and density of limit interfaces. Geom. Funct. Anal. 29, 382–410 (2019). https://doi.org/10.1007/s00039-019-00489-1

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  • DOI: https://doi.org/10.1007/s00039-019-00489-1

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