Volume estimates for Alexandrov spaces with convex boundaries

Jian Ge

Vol. 314 (2021), No. 2, 269–281

DOI: 10.2140/pjm.2021.314.269

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