#### 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
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