Deformations of Totally Geodesic Foliations and Minimal Surfaces in Negatively Curved 3-Manifolds

TL;DR

This paper studies how minimal surface foliations in negatively curved 3-manifolds deform from totally geodesic foliations, providing constructions, conditions for existence, and examples where such foliations cannot exist.

Contribution

It constructs deformations of totally geodesic foliations into minimal surface foliations under curvature constraints and identifies obstructions to their existence.

Findings

Constructed foliations of Grassmann bundles with minimal surfaces.

Identified curvature conditions for the persistence of foliations.

Provided examples where such foliations cannot exist.

Abstract

Let $g_t$ 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 $(M,g_t)$. These foliations are deformations of the foliation of $Gr_2(M)$ by (lifts of) totally geodesic planes projected down from the universal cover $\mathbb{H}^3$. 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.