Volume estimates for Alexandrov Spaces with convex boundaries
Differential Geometry
2020-10-23 v1 Metric Geometry
Abstract
In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space 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 also can be applied to Riemannian manifolds with non-smooth boundary, which generalizes Heintze and Karcher's classical volume comparison theorem. Our main tool is the gradient flow of semi-concave functions.
Cite
@article{arxiv.2010.11760,
title = {Volume estimates for Alexandrov Spaces with convex boundaries},
author = {Jian Ge},
journal= {arXiv preprint arXiv:2010.11760},
year = {2020}
}