Volume comparison on finite-volume hyperbolic 3-manifolds

2025/09/03 by Jiang, Ruojing, Pallete, Franco Vargas
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2509.03566

Abstract

On finite-volume hyperbolic 3-manifolds, we compare volumes of different metrics using the exponential convergence of Ricci-DeTurck flow toward the hyperbolic metric h₀. We prove that among metrics with scalar curvature bounded below by -6, h₀ minimizes the volume. Moreover, for metrics that are either uniformly C²-close to h₀ or asymptotically cusped of order at least two, equality holds if and only if the metric is isometric to h₀.

Citations

• On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume
• Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds
• Normalized Ricci flows and conformally compact Einstein metrics
• Stability of hyperbolic manifolds with cusps under Ricci flow
• Asymptotic Stability of the Cross Curvature Flow at a Hyperbolic Metric
• Lower bounds on volumes of hyperbolic Haken 3-manifolds

Related

• On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume
• Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds
• Volume comparison with respect to scalar curvature
• The volume comparison of symmetric spaces of non-compact type of rank 1
• Scalar curvature and volume entropy of hyperbolic 3-manifolds