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Applications of Min–Max Methods to Geometry

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Geometric Analysis

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2263))

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Abstract

The existence of minimal surfaces in closed manifolds is a classical subject with a long history. This chapter presents some recent advances on the subject, motivated by Yau’s conjecture concerning the existence of infinitely-many ones. The main tools used here are a combination of techniques from Geometric Measure Theory and Minimal methods. The conjecture is proved for a large class of metrics and, via the concept of volume spectrum, a density result is also derived.

The first author is partly supported by NSF-DMS-1811840. The second author is partly supported by NSF DMS-1710846 and by a Simons Investigator Grant.

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Author information

Authors and Affiliations

  1. Institute for Advanced Study, Princeton University Princeton, Princeton, NJ, USA

    Fernando C. Marques

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

    André Neves

Authors
  1. Fernando C. Marques
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  2. André Neves
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Correspondence to André Neves .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA

    Matthew J. Gursky

  2. Scuola Normale Superiore, Pisa, Italy

    Andrea Malchiodi

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Marques, F.C., Neves, A. (2020). Applications of Min–Max Methods to Geometry. In: Gursky, M., Malchiodi, A. (eds) Geometric Analysis . Lecture Notes in Mathematics(), vol 2263. Springer, Cham. https://doi.org/10.1007/978-3-030-53725-8_2

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