For finite-volume hyperbolic 3-manifolds, the hyperbolic metric uniquely minimizes minimal surface entropy among sectional curvature at most -1 metrics and uniquely maximizes it among scalar curvature at least -6 metrics under rigidity or C0-closeness hypotheses.
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Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds
For finite-volume hyperbolic 3-manifolds, the hyperbolic metric uniquely minimizes minimal surface entropy among sectional curvature at most -1 metrics and uniquely maximizes it among scalar curvature at least -6 metrics under rigidity or C0-closeness hypotheses.