2025/01/21 by Nguyen, Manh-Tien, Schlenker, Jean-Marc, Seppi, Andrea
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2501.12277
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×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.