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Existence of hypersurfaces with prescribed mean curvature I - Generic min-max

Xin Zhou University of California Santa Barbara Jonathan J. Zhu Princeton University

arXiv subject: Differential Geometry (math.DG) mathscidoc:2005.53002

Cambridge Journal of Mathematics, 8, (2), 311–362, 2020
We prove that, for a generic set of smooth prescription functions h on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature h. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersurface of multiplicity one. More precisely, we show that our previous min-max theory, developed for constant mean curvature hypersurfaces, can be extended to construct min-max prescribed mean curvature hypersurfaces for certain classes of prescription function, including a generic set of smooth functions, and all nonzero analytic functions. In particular we do not need to assume that h has a sign.
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  • https://www.intlpress.com/site/pub/pages/journals/items/cjm/content/vols/0008/0002/a002/index.php
@inproceedings{xin2020existence,
  title={Existence of hypersurfaces with prescribed mean curvature I - Generic min-max},
  author={Xin Zhou, and Jonathan J. Zhu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200510223513299478668},
  booktitle={Cambridge Journal of Mathematics},
  volume={8},
  number={2},
  pages={311–362},
  year={2020},
}
Xin Zhou, and Jonathan J. Zhu. Existence of hypersurfaces with prescribed mean curvature I - Generic min-max. 2020. Vol. 8. In Cambridge Journal of Mathematics. pp.311–362. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200510223513299478668.
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