Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces

Diptaishik Choudhury (1) , Filippo Mazzoli (2) , Andrea Seppi (3)

1 University of Luxembourg [Luxembourg]
2 University of Virginia
3 IF - Institut Fourier

Even though it is known that there exist quasi-Fuchsian hyperbolic three-manifolds that do not admit any monotone foliation by constant mean curvature (CMC) surfaces, a conjecture due to Thurston asserts the existence of CMC foliations for all almost-Fuchsian manifolds, namely those quasi-Fuchsian manifolds that contain a closed minimal surface with principal curvatures in (-1,1). In this paper we prove that there exists a (unique) monotone CMC foliation for all quasi-Fuchsian manifolds that lie in a sufficiently small neighborhood of the Fuchsian locus.

Domaines: Géométrie différentielle [math.DG]

Diptaishik Choudhury, Filippo Mazzoli, Andrea Seppi. Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces. Mathematische Annalen, 2023, ⟨10.1007/s00208-023-02625-7⟩. ⟨hal-03745084⟩