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Min-max theory for free boundary minimal hypersurfaces I-regularity theory

Martin Man-chun Li Chinese University of Hong Kong Xin Zhou University of California Santa Barbara

Differential Geometry Geometric Analysis and Geometric Topology mathscidoc:1910.43013

arXiv preprint arXiv:1611.02612, 2016.11
In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, we prove the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves shows that for any compact Riemannian manifolds with nonnegative Ricci curvature and convex boundary, there exist infinitely many embedded minimal hypersurfaces with free boundary which are properly embedded.
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  • to appear in Journal of Differential Geometry
@inproceedings{martin2016min-max,
  title={Min-max theory for free boundary minimal hypersurfaces I-regularity theory},
  author={Martin Man-chun Li, and Xin Zhou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020105634454611542},
  booktitle={arXiv preprint arXiv:1611.02612},
  year={2016},
}
Martin Man-chun Li, and Xin Zhou. Min-max theory for free boundary minimal hypersurfaces I-regularity theory. 2016. In arXiv preprint arXiv:1611.02612. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020105634454611542.
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