Minimal Surfaces in Hyperbolizable 3-Manifolds

Abstract

In the first part of this thesis we study the extent to which statements about negatively curved manifolds hold when geodesics, which are one-dimensional minimal surfaces, are replaced by two-dimensional minimal surfaces. For a certain class of negatively curved 3-manifolds $M$ we construct minimal surfaces whose tangential lifts foliate the Grassmann bundle of tangent 2-planes to $M$. These foliations make it possible to use homogeneous dynamics to study how minimal surfaces in variable negative curvature are distributed in the ambient space. On the other hand, we construct negatively curved $M$ for which there cannot exist a foliation as above. In the second part (joint with Zeno Huang), we find new estimates for several measures of the complexity of hyperbolic 3-manifolds homeomorphic to $\Sigma \times \mathbb{R}$ in terms of the geometry of the minimal surfaces that they contain. This leads to a ``gap" theorem for the geometry of minimal surfaces in hyperbolic 3-manifolds that fiber over the circle.

In the final section (joint with Andre Neves), we prove new lower bounds for the areas of certain surfaces in a hyperbolizable Riemannian 3-manifold M under a lower bound on the scalar curvature of M. Our bounds are sharp and improve the best bounds known previously by a factor of three. We also prove sharp lower bounds for an entropy functional based on asymptotic counts of minimal surfaces recently introduced by Calegari-Marques-Neves and verify a conjecture of Gromov on average area-distortion of maps from M to a hyperbolic 3-manifold.