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

TL;DR

The paper extends volume comparison results to rank-1 symmetric spaces of non-compact type under scalar curvature bounds by employing the normalized Ricci–DeTurck flow. It establishes monotonicity and nonnegativity of the relative volume $V_{g_0}(g)$ for small, exponentially weighted perturbations with $R\ge R_0$, and proves a rigidity theorem when equality is achieved, showing $g$ is a pullback of $g_0$ by a diffeomorphism. The approach relies on explicit root-theoretic constants $\gamma$ and $\tau_0$, long-time decay and spatial decay estimates for NRDF on Cartan–Hadamard manifolds, and a detailed analysis of scalar curvature evolution. Collectively, these results provide a noncompact analog of Schoen’s volume conjecture for rank-1 spaces and extend the classical volume comparison framework to non-compact symmetric spaces via a dynamical, PDE-based method.

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.

Theorems & Definitions (31)

• Conjecture 1.1: Schoen's Conjecture Sc1989
• Theorem 1.1
• Remark 1.1
• Remark 2.1
• Remark 2.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3