Mean curvature flow in homology and foliations of hyperbolic $3$-manifolds

TL;DR

This paper investigates the behavior of mean curvature flow in hyperbolic 3-manifolds, establishing convergence to minimal surfaces, and constructs foliations with minimal or non-vanishing mean curvature leaves, revealing new geometric structures.

Contribution

It demonstrates convergence of mean curvature flow to minimal hypersurfaces and constructs smooth foliations in hyperbolic 3-manifolds with minimal or non-vanishing mean curvature leaves, linking flow dynamics with manifold topology.

Findings

Flow converges smoothly to minimal hypersurfaces

Existence of foliations with minimal or non-vanishing mean curvature leaves

Outermost minimal surfaces in quasi-Fuchsian ends

Abstract

We study global aspects of the mean curvature flow of non-separating hypersurfaces $S$ in closed manifolds. For instance, if $S$ has non-vanishing mean curvature, we show its level set flow converges smoothly towards an embedded minimal hypersurface $\Gamma$. We prove a similar result for the flow with surgery in dimension 2. As an application we show the existence of monotone incompressible isotopies in manifolds with negative curvature. Combining this result with min-max theory, we show that quasi-Fuchsian and hyperbolic $3$-manifolds fibered over $\mathrm{S}^1$ admit smooth entire foliations whose leaves are either minimal or have non-vanishing mean curvature. We also conclude the existence of outermost minimal surfaces for quasi-Fuchsian ends and study their continuity with respect to variations of the quasi-Fuchsian metric.