Min-max theory for constant mean curvature hypersurfaces

2017/07/25 by Xin Zhou, Zhou, Xin, Jonathan J. Zhu +1 · 2 citations
Mathematics · #49J35 #53A10 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1707.08012

openalex publication_date 2017/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The existence of surfaces of constant mean curvature with free boundaries
• Entropy of closed surfaces and min-max theory

Cited by

• Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics
• Area bounds for constant mean curvature surfaces in hyperbolic 3-manifolds

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