1. Weakly almost-Fuchsian manifolds are nearly-Fuchsian
- Author
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Nguyen, Manh-Tien, Schlenker, Jean-Marc, and Seppi, Andrea
- Subjects
Mathematics - Differential Geometry - 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., Comment: 18 pages
- Published
- 2025