On any closed hyperbolizable 3-manifold, we find a sharp relation between the minimal surface entropy (introduced by Calegari-Marques-Neves) and the average area ratio (introduced by Gromov), and we show that, among metrics g with scalar curvature greater than or equal to −6, the former is maximized by the hyperbolic metric. One corollary is to solve a conjecture of Gromov regarding the average area ratio.Our proofs use Ricci flow with surgery and laminar measures invariant under a PSL(2, R)-action.

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Let M be a closed hyperbolic 3-manifold that admits no infinitesimal conformally-flat deformations. Examples of such manifolds were constructed by Kapovich. Then if g is a Riemannian metric on M with scalar curvature greater than or equal to −6, we find lower bounds for the areas of stable immersed minimal surfaces Σ in M . Our bounds improve the closer Σ is to being homotopic to a totally geodesic surface in the hyperbolic metric.We also consider a functional introduced by Calegari-Marques-Neves that is defined by an asymptotic count of minimal surfaces in (M, g). We show this functional to be uniquely maximized, over all metrics of scalar curvature greater than or equal to −6, by the hyperbolic metric. Our proofs use the Ricci flow with surgery.

An almost Fuchsian manifold is a hyperbolic three-manifold of the type S × R which admits a closed minimal surface (homeomorphic to S) with the maximum principal curvature λ 0 < 1, while a weakly almost Fuchsian manifold is of the same type but it admits a closed minimal surface with λ 0 ≤ 1. We first prove that any weakly almost Fuchsian manifold is in fact quasi-Fuchsian, and we construct a Canary-Storm type compactification for the weakly almost Fuchsian space. We apply these results to prove uniform upper bounds on the volume of the convex core and Hausdorff dimension for the limit sets of weakly almost Fuchsian manifolds and to give examples of quasi-Fuchsian manifolds which admit unique minimal surfaces (resp. stable minimal surfaces) without being almost Fuchsian (resp. weakly almost Fuchsian). We also show that for every g there is an ǫ depending only on g such that if a closed hyperbolic three-manifold fibers over the circle with fiber a surface of genus g, then any embedded minimal surface isotopic to the fiber has the maximum principal curvature λ 0 larger than 1 + ǫ.

Let gt be a smooth 1-parameter family of negatively curved metrics on a closed hyperbolic 3-manifold M starting at the hyperbolic metric. We construct foliations of the Grassmann bundle Gr 2 (M ) of tangent 2-planes whose leaves are (lifts of) minimal surfaces in (M, gt). These foliations are deformations of the foliation of Gr 2 (M ) by (lifts of) totally geodesic planes projected down from the universal cover H 3 . Our construction continues to work as long as the sum of the squares of the principal curvatures of the (projections to M ) of the leaves remains pointwise smaller in magnitude than the ambient Ricci curvature in the normal direction. In the second part of the paper we give some applications and construct negatively curved metrics for which Gr 2 (M ) cannot admit a foliation as above.