Sharp bounds and rigidity for volumes of boundaries of Alexandrov spaces

TL;DR

This paper establishes precise volume bounds for the boundaries of Alexandrov spaces with specific curvature, dimension, and radius constraints, providing a complete classification of cases where these bounds are tight and exploring near-rigidity scenarios.

Contribution

It introduces sharp volume bounds for Alexandrov space boundaries and fully characterizes the rigidity cases, extending results to smooth manifolds with boundary.

Findings

Sharp volume bounds for boundaries of Alexandrov spaces

Complete classification of rigidity cases

Analysis of almost rigidity scenarios

Abstract

We obtain sharp volume bounds on the boundaries of Alexandrov spaces with given lower curvature bound, dimension, and radius. We also completely classify the rigidity case and analyze almost rigidity. Our results are new even for smooth manifolds with boundary.