# Min–max theory for constant mean curvature hypersurfaces

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## Abstract

In this paper, we develop a min–max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature *c*. Moreover, if *c* is nonzero then our min–max solution always has multiplicity one.

## Notes

### Acknowledgements

Both authors are grateful to Prof. Shing-Tung Yau for suggesting this problem and for his generous support. X. Zhou would also like to thank Prof. Richard Schoen and Prof. Neshan Wickramasekera for valuable comments. J. Zhu would also like to thank Prof. William Minicozzi for his invaluable guidance and encouragement. X. Zhou is partially supported by NSF grant DMS-1704393 and DMS-1811293. J. Zhu is partially supported by NSF grant DMS-1607871. Finally, both authors would like to thank the anonymous referees for their comments.

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