On almost-Fuchsian manifolds (Q2847121)

An almost-Fuchsian manifold is, by definition, a quasi-Fuchsian manifold containing a closed incompressible surface with principal curvatures taking values in \((-1,1)\). They form a subspace, of complex dimension \(6g-6\), in the moduli space of quasi-Fuchsian manifolds (here \(g\) is the genus of any closed incompressible surface in the manifold).NEWLINENEWLINEThe pinching condition of curvatures implies the existence of a unique minimal surface by a work of \textit{K. K. Uhlenbeck} [Ann. Math. Stud. 103, 147--168 (1983; Zbl 0529.53007)].NEWLINENEWLINEThe paper under review discusses two quantitative properties of almost-Fuchsian manifolds and exhibits a quasi-Fuchsian manifold with a new feature.NEWLINENEWLINEThe first two results estimate the hyperbolic volume of a convex core and the Hausdorff dimension of the limit set of the associated Kleinian group in terms of the maximum \(\lambda_0\) of the principal curvatures of the unique minimal surface in an almost-Fuchsian manifold. More precisely, the authors prove that the volume of a convex core is bounded from above by \(2\pi(2g-2)(2\lambda_0+4/3 \lambda_0^3+(\text{higher terms}))\), and the Hausdorff dimension of the limit set is bounded, also from above, by \(1+\lambda_0^2\). To describe the last result we recall that the almost-Fuchsian manifolds can be used to produce a foliation by surfaces of constant mean curvature. A natural question, due to Thurston, is to what extent a quasi-Fuchsian manifold admits a foliation by surfaces of constant mean curvatures. The authors exhibit a quasi-Fuchsian manifold which does not admit a foliation by constant mean curvature surfaces.