Volume comparison on finite-volume hyperbolic 3-manifolds

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 $h_0$. We prove that among metrics with scalar curvature bounded below by $-6$, $h_0$ minimizes the volume. Moreover, for metrics that are either uniformly $C^2$-close to $h_0$ or asymptotically cusped of order at least two, equality holds if and only if the metric is isometric to $h_0$.