Weakly almost-Fuchsian manifolds are nearly-Fuchsian

TL;DR

This paper proves that certain hyperbolic 3-manifolds with minimal surfaces of bounded principal curvatures are close to Fuchsian, establishing new links between minimal surface properties and the quasi-Fuchsian condition.

Contribution

It demonstrates that hyperbolic 3-manifolds with minimal surfaces of principal curvatures in [-1,1] contain nearby surfaces with curvatures in (-1,1), and shows that some quasi-Fuchsian manifolds lack minimal surfaces with these curvatures, answering open questions.

Findings

Hyperbolic 3-manifolds with minimal surfaces of principal curvatures in [-1,1] contain nearby non-minimal surfaces with curvatures in (-1,1).

Complete hyperbolic 3-manifolds homeomorphic to S×R with such surfaces are quasi-Fuchsian.

Existence of quasi-Fuchsian manifolds with surfaces of principal curvatures in (-1,1) but no minimal such surfaces.

Abstract

We show that a hyperbolic three-manifold $M$ containing a closed minimal surface with principal curvatures in $[-1,1]$ also contains nearby (non-minimal) surfaces with principal curvatures in $(-1,1)$. When $M$ is complete and homeomorphic to $S\times\mathbb{R}$, for $S$ a closed surface, this implies that $M$ is quasi-Fuchsian, answering a question left open from Uhlenbeck's 1983 seminal paper. Additionally, our result implies that there exist (many) quasi-Fuchsian manifolds that contain a closed surface with principal curvatures in $(-1,1)$, but no closed minimal surface with principal curvatures in $(-1,1)$, disproving a conjecture from the 2000s.