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arXiv · 2508.08674

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

Jiaqi Chen·Yufei Shan·Yinghui Ye
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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.

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Jiaqi Chen, Yufei Shan, Yinghui Ye. 2025-08-12. The volume comparison of symmetric spaces of non-compact type of rank 1. https://arxiv.org/abs/2508.08674

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