Comparing the Morse index and the first Betti number of minimal hypersurfaces

2016/01/29 by Lucas Ambrozio, Ambrozio, Lucas, Alessandro Carlotto +3 · 3 citations
Mathematics · #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.1601.08152

openalex publication_date 2016/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By extending and generalising previous work by Ros and Savo, we describe a method to show that the Morse index of every closed minimal hypersurface on certain positively curved ambient manifolds is bounded from below by a linear function of its first Betti number. The technique is flexible enough to prove that such a relation between the index and the topology of minimal hypersurfaces holds, for example, on all compact rank one symmetric spaces, on products of the circle with spheres of arbitrary dimension and on suitably pinched submanifolds of the Euclidean spaces. These results confirm a general conjecture due to Schoen and Marques-Neves for a wide class of ambient spaces.

Citations

• Minimal Varieties in Riemannian Manifolds
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

Cited by

• Regularity of Cohomogeneity two equivariant isotopy minimization problems and minimal hypersurfaces with large first Betti number on spheres
• Minimal hypersurfaces of Morse index one
• Ricci curvature and minimal hypersurfaces with large Betti numbers

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