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Equidistribution of minimal hypersurfaces for generic metrics

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Abstract

For almost all Riemannian metrics (in the \(C^\infty \) Baire sense) on a closed manifold \(M^{n+1}\), \(3\le (n+1)\le 7\), we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in M. This gives a quantitative version of the main result of Irie et al. (Ann Math 187(3):963–972, 2018), that established density of minimal hypersurfaces for generic metrics. As in Irie et al. (2018), the main tool is the Weyl Law for the Volume Spectrum proven by Liokumovich et al. (Ann Math 187(3):933–961, 2018).

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Authors and Affiliations

  1. Princeton University, Fine Hall, Princeton, NJ, 08544, USA

    Fernando C. Marques & Antoine Song

  2. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    André Neves

  3. Imperial College London, Huxley Building 180 Queen’s Gate, London, SW7 2RH, UK

    André Neves

Authors
  1. Fernando C. Marques
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  2. André Neves
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  3. Antoine Song
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Correspondence to André Neves.

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The first author is partly supported by NSF-DMS-1509027 and NSF DMS-1311795. The second author is partly supported by NSF DMS-1710846 and EPSRC Programme Grant EP/K00865X/1. The third author is supported by NSF-DMS-1509027.

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Marques, F.C., Neves, A. & Song, A. Equidistribution of minimal hypersurfaces for generic metrics. Invent. math. 216, 421–443 (2019). https://doi.org/10.1007/s00222-018-00850-5

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  • DOI: https://doi.org/10.1007/s00222-018-00850-5

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