Volume estimates for tubes around submanifolds using integral curvature\n bounds

2018/10/03 by Yousef K. Chahine, Chahine, Yousef K. · 1 citation
Mathematics · Medicine · #53C20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Pelvic and Acetabular Injuries #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1810.01935

openalex publication_date 2018/10/03 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28

Abstract

We generalize an inequality of E. Heintze and H. Karcher [8] for the volume\nof tubes around minimal submanifolds to an inequality based on integral bounds\nfor k-Ricci curvature. Even in the case of a pointwise bound, this\ngeneralizes the classical inequality by replacing a sectional curvature bound\nwith a k-Ricci bound. This work is motivated by the estimates of\nPetersen-Shteingold-Wei for the volume of tubes around a geodesic [12] and\ngeneralizes their result. Using similar ideas we also prove a Hessian\ncomparison theorem for k-Ricci curvature which generalizes the usual Hessian\nand Laplacian comparison for distance functions from a point and give several\napplications.\n

Cited by

• Fenchel-Willmore inequality for submanifolds in manifolds with non-negative k-Ricci curvature

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