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Min-max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications

Qiang Guang Australian National University Martin Man-chun Li Chinese University of Hong Kong Zhichao Wang Max-Planck Institute for Mathematics Xin Zhou University of California Santa Barbara

Differential Geometry Geometric Analysis and Geometric Topology mathscidoc:2001.10005

arXiv preprint arXiv:1907.12064, 2019.7
For any smooth Riemannian metric on an (n+1)-dimensional compact manifold with boundary (M,∂M) where 3≤(n+1)≤7, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates to prove that for almost every (in the C-infinity Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M. If ∂M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.
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@inproceedings{qiang2019min-max,
  title={Min-max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications},
  author={Qiang Guang, Martin Man-chun Li, Zhichao Wang, and Xin Zhou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200103164337831335619},
  booktitle={arXiv preprint  arXiv:1907.12064},
  year={2019},
}
Qiang Guang, Martin Man-chun Li, Zhichao Wang, and Xin Zhou. Min-max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications. 2019. In arXiv preprint arXiv:1907.12064. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200103164337831335619.
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