Min–max theory for constant mean curvature hypersurfaces

By Xin Zhou, Jonathan J. Zhu. Inventiones Mathematicae, 2017

77 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4c87a3709d010a86744b106688192b89ee8b3928

How researchers see this paper

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In Applied Mathematics

1. Jonathan J. Zhu
2. Xin Zhou
3. Otis Chodosh
4. Marco A. M. Guaraco
5. Antoine Song
6. F. C. Marques
7. A. Neves
8. P. Gaspar

In Geometry and Topology

1. Jonathan J. Zhu
2. Xin Zhou
3. Marco A. M. Guaraco
4. Antoine Song
5. F. C. Marques
6. A. Neves
7. P. Gaspar

Citations in the corpus

• On the Multiplicity One Conjecture in min-max theory (2019, unknown): “Here we present an adaption to multi-parameter families of the min-max theory for hypersurfaces with prescribed mean curvature (abbreviated as PMC) established by the author with Zhu [51, 52].”
• Morse index of multiplicity one min-max minimal hypersurfaces (2015, unknown): “It would be interesting to know if similar index bounds hold in other min-max constructions (e.g. ChambersLiokumovich [8], De Lellis-Ramic [11], De Lellis-Tasnady [12], Li-Zhou [27], Montezuma [39]-[40], Zhou-Zhu [64]).”
• Existence of hypersurfaces with prescribed mean curvature I – generic min-max (2018, methodology): “Instead, we use another interpolation method in Claim 2 as developed in [65].”