Vol. 314, No. 2, 2021

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Volume estimates for Alexandrov spaces with convex boundaries

Jian Ge

Vol. 314 (2021), No. 2, 269–281
Abstract

We estimate the upper bound of volume of a closed positively or nonnegatively curved Alexandrov space X with strictly convex boundary. We also discuss the equality case. In particular, the boundary conjecture holds when the volume upper bound is achieved. Our theorem can also be applied to Riemannian manifolds with nonsmooth boundary, which generalizes Heintze and Karcher’s classical volume comparison theorem. Our main tool is the gradient flow of semiconcave functions.

Keywords
Alexandrov space, volume comparison, convex boundary, gradient flow
Mathematical Subject Classification
Primary: 53C23
Milestones
Received: 1 November 2020
Revised: 8 July 2021
Accepted: 14 August 2021
Published: 10 November 2021
Authors
Jian Ge
Beijing Normal University
Beijing
China