On the Multiplicity One Conjecture in Min-max theory

Xin Zhou University of California Santa Barbara

arXiv subject: Differential Geometry (math.DG) mathscidoc:2005.53003

2019
We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each volume spectrum is realized by the min-max value of certain relative homotopy class of sweepouts of boundaries of Caccioppoli sets. The main result follows by approximating such min-max value using the min-max theory for hypersurfaces with prescribed mean curvature established by the author with Zhu.
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  • https://arxiv.org/abs/1901.01173
@inproceedings{xin2019on,
  title={On the Multiplicity One Conjecture in Min-max theory},
  author={Xin Zhou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200510223732049887669},
  year={2019},
}
Xin Zhou. On the Multiplicity One Conjecture in Min-max theory. 2019. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200510223732049887669.
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