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

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.