Sagitta, Lenses, and Maximal Volume

2014/08/23 by Curtis Pro, Pro, Curtis
Mathematics · #53C20 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1408.5534

openalex publication_date 2014/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a characterization of critical points that allows us to define a metric invariant on all Riemannian manifolds M with a lower sectional curvature bound and an upper radius bound. We show there is a uniform upper volume bound for all such manifolds with an upper bound on this invariant. We generalize results by Grove and Petersen and by Sill, Wilhelm, and the author by showing any such M that has volume sufficiently close to this upper bound is diffeomorphic to the standard sphere Sⁿ or a standard lens space Sⁿ/ℤ_m where m∈\2,3,…\ is no larger than an a priori constant.

Related

• The Diffeomorphism Type of Manifolds with Almost Maximal Volume
• Critical metrics of the volume functional on complete manifolds
• Critical metrics and curvature of metrics with unit volume or unit area of the boundary
• Rigidity theorems for the area widths of Riemannian manifolds
• Isoperimetric inequality and Weitzenböck type formula for critical metrics of the volume