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arXiv · 2501.12277

Weakly almost-Fuchsian manifolds are nearly-Fuchsian

Manh-Tien Nguyen·Jean-Marc Schlenker·Andrea Seppi
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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.

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Manh-Tien Nguyen, Jean-Marc Schlenker, Andrea Seppi. 2025-01-21. Weakly almost-Fuchsian manifolds are nearly-Fuchsian. https://arxiv.org/abs/2501.12277

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