By a result of Mazzeo-Pacard, every hyperbolic quasi-Fuchsian manifold admits a compact subset whose complement is foliated by constant mean curvature (CMC) surfaces. However, there exist quasi-Fuchsian manifolds that do not admit a global CMC foliation. A conjecture due to Thurston asserts that every almost-Fuchsian manifold has such a global CMC foliation.

In this talk I will discuss a partial result in this direction, obtained in a joint work with Diptaishik Choudhury and Filippo Mazzoli: every quasi-Fuchsian manifold in a neighbourhood of the Fuchsian locus is (uniquely and monotonically) foliated by CMC surfaces.