Volume estimates for Alexandrov Spaces with convex boundaries

TL;DR

This paper provides volume upper bounds for Alexandrov spaces with convex boundaries, generalizing classical theorems to non-smooth boundary cases and analyzing equality conditions.

Contribution

It introduces new volume estimates for Alexandrov spaces with convex boundaries and extends classical volume comparison results to non-smooth boundary settings.

Findings

Volume upper bounds for Alexandrov spaces with convex boundaries.

Validation of the Boundary Conjecture when bounds are achieved.

Application of gradient flow of semi-concave functions in the proof.

Abstract

In this note, we estimate the upper bound of volume of 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 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.