11 problems

Let $S$ be a closed orientable surface of genus $k\geq 2$. For a quasi-Fuchsian hyperbolic 3-manifold, let its convex core be the minimal compact totally convex subset, whose bound…

McMullen's conjecture. The convex core $C(N)$ does not contain an embedded ball of radius $L$, where $L$ depends on the number of generators of $\pi_1(N)$.

Unique global minimum conjecture. For every $M\in\mathcal{I}(N)\setminus\{M_0\}$,

Let $S$ be a closed orientable surface of genus $k\geq 2$. A GHMC AdS (2+1)-spacetime is a globally hyperbolic maximal Cauchy compact anti-de Sitter (2+1)-spacetime, homeomorphic t…

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…

A convex core is the quotient of the convex hull of the limit set of a convex cocompact hyperbolic 3-manifold; its boundary carries an induced hyperbolic metric and a pleating lami…

Let $N$ be a hyperbolizable 3-manifold, let $CC(N)$ be its deformation space of convex cocompact structures, and let $V$ be the Weil–Petersson gradient vector field of renormalized…

Let $M$ be a geometrically finite hyperbolic 3-manifold, meaning that its convex core has finite volume. Its convex core is the intersection of all closed totally convex subsets of…

Mess's conjecture. Conversely, any pair

Mess's conjecture. There is a unique maximal globally hyperbolic $\mathbb{A}\mathbbm{d}\mathbb{S}^3$ manifold $N$ such that the induced metric on the boundary of the convex core of…

Induced-metric uniqueness conjecture. The analog for GHMC AdS manifolds of the theorem asserting existence of a quasifuchsian cone-manifold with prescribed induced metric on the bo…