The hyperbolic metric minimizes volume among all metrics with scalar curvature at least -6 on finite-volume hyperbolic 3-manifolds, with rigidity when the metric is C^2-close or asymptotically cusped.
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Volume comparison on finite-volume hyperbolic 3-manifolds
The hyperbolic metric minimizes volume among all metrics with scalar curvature at least -6 on finite-volume hyperbolic 3-manifolds, with rigidity when the metric is C^2-close or asymptotically cusped.