Weakly almost-Fuchsian manifolds are nearly-Fuchsian

TL;DR

The paper proves that a hyperbolic 3-manifold $M$ containing a closed minimal surface with principal curvatures in $[-1,1]$ also contains nearby non-minimal surfaces with principal curvatures in $(-1,1)$, yielding a local deformation into a nearly-Fuchsian geometry. It shows that, for complete $M$ homeomorphic to $S\times\mathbb{R}$, such manifolds are quasi-Fuchsian, answering a long-standing question of Uhlenbeck. It further demonstrates that weakly almost-Fuchsian manifolds are nearly-Fuchsian and provides a counterexample to the conjecture that nearly-Fuchsian implies almost-Fuchsian, while also establishing a universal ε-converse: quasi-Fuchsian manifolds with a surface of curvatures in $(-ε,ε)$ are almost-Fuchsian. The approach blends normal variation analysis of minimal surfaces, the cosh-Gordon equation, and half-translation structures to control principal curvatures under deformations. These results refine the deformation theory of convex surfaces in hyperbolic 3-manifolds and have implications for the structure of the quasi-Fuchsian deformation space and related geometric conjectures.

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.

Theorems & Definitions (15)

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