5 problems

Nearly-Fuchsian implies almost-Fuchsian conjecture. If a quasi-Fuchsian manifold $M$ contains a closed surface with principal curvatures in $(-1,1)$, then it contains a closed mini…

Thurston's conjecture. For any two hyperbolic metrics $h^{\pm}$ on $S$ up to isotopy, there exists a unique quasi-Fuchsian manifold $Q$, up to isotopy, that induces $h^{\pm}$ on…

Let $S$ be a closed surface, let $\mathcal{FMF}(S)$ denote the space of pairs of measured foliations on $S$, and let $t>0$. The current result concerns measured foliations at infin…

Thurston's conjecture. The map $\Psi$ should be a parametrization of the space of quasi-Fuchsian manifolds; equivalently, it should be a homeomorphism.

Let $X$ be the conformal surface at infinity of a geometrically finite end of a hyperbolic $3$-manifold, and let $I_k$ and $I\!I_k$ denote the first and second fundamental forms of…