On the Multiplicity One Conjecture in Min-max theory

2019/01/04 by Xin Zhou, Zhou, Xin · 8 citations
Mathematics · #Analysis of PDEs (math.AP) #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.1901.01173

openalex publication_date 2019/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each volume spectrum is realized by the min-max value of certain relative homotopy class of sweepouts of boundaries of Caccioppoli sets. The main result follows by approximating such min-max value using the min-max theory for hypersurfaces with prescribed mean curvature established by the author with Zhu.

Citations

• Curvature estimates for minimal hypersurfaces
• On the Fundamental Group of a Compact Minimal Submanifold
• Minimal Varieties in Riemannian Manifolds
• Existence of hypersurfaces with prescribed mean curvature I - Generic min-max

Cited by

• Regularity of Cohomogeneity two equivariant isotopy minimization problems and minimal hypersurfaces with large first Betti number on spheres
• Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics
• Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature
• Varifold convergence of free boundary Allen--Cahn equation
• Local Variational Properties of Non-degenerate Free Boundary Minimal Hypersurfaces
• Mountain Pass Critical Points of the Volume Constrained Area Functional
• Ricci curvature and minimal hypersurfaces with large Betti numbers
• When can minimal hypersurfaces be connected by mean curvature flow?

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• Min-max minimal hypersurfaces with higher multiplicity
• Multiplicity one for min-max theory in compact manifolds with boundary and its applications
• Min-max hypersurface in manifold of positive Ricci curvature
• Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature
• Generic existence of multiplicity-1 minmax minimal hypersurfaces via Allen--Cahn