The volume comparison of symmetric spaces of non-compact type of rank 1

TL;DR

This paper extends volume comparison theorems to rank 1 symmetric spaces of non-compact type using Ricci flow techniques, providing new insights into geometric rigidity under scalar curvature conditions.

Contribution

It generalizes existing volume comparison results to a broader class of symmetric spaces and introduces a novel flow-based approach for analyzing volume functionals.

Findings

Established a volume comparison theorem for rank 1 symmetric spaces

Proved a rigidity result under scalar curvature conditions

Used normalized Ricci--DeTurck flow to analyze volume behavior

Abstract

Motivated by Schoen's conjecture on the volume functional for closed hyperbolic manifolds, we generalize the volume comparison theorem of Hu, Ji, and Shi and establish a volume comparison theorem for rank 1 symmetric spaces of non-compact type under a scalar curvature condition. Furthermore, we prove a rigidity result. Our proof uses the normalized Ricci--DeTurck flow to analyze the asymptotic behavior of the volume functional and to derive monotonicity properties. This extends the classical volume comparison framework to symmetric spaces of non-compact type.