16 problems

Intrinsic metric homeomorphism conjecture. The map $\mathcal{I}_-^s$ is a homeomorphism.

Goldman's deformation conjecture. Any small deformation of a standard cocompact discontinuous group preserves proper discontinuity.

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…

A globally hyperbolic compact anti-de Sitter (2+1)-spacetime is a spacetime with the stated global hyperbolicity and compactness properties, and with a spacelike convex boundary. I…

Mess's conjecture. Conversely, any pair

Barbot's component conjecture. For $p\geq 3$, any $\mathbb{H}^{p,1}$-convex cocompact representation of $\Gamma$ lies in $\mathcal{T}_{\rho_0}$.

Let $n=2d$ be even, let $\operatorname{AdS}^{1,2d}$ be anti-de Sitter space, and let $U(1,d)\subset\operatorname{SO}_0(2,2d)$ act on it. A closed Lorentzian manifold of constant ne…

Let an MGHC anti-de Sitter spacetime be a maximal globally hyperbolic spatially compact spacetime modeled on anti-de Sitter space, and let $\Lambda$ be its associated achronal subs…

Let an asymptotically anti-de Sitter (aAdS) black hole be a black-hole spacetime with asymptotically anti-de Sitter geometry, and let its quasinormal frequencies (QNFs) be the comp…

Let $n=2k+1>4$ be odd, and let $\Gamma$ be a discrete subgroup of $\operatorname{SO}_0(n-1,2)$ acting freely, properly discontinuously and cocompactly on anti-de Sitter space…

A closed anti-de Sitter 3-manifold is a closed $3$-manifold equipped with a Lorentz metric of constant negative sectional curvature; it is standard if its anti-de Sitter structure…

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…

AdS convex-boundary conjecture. There exists a unique globally hyperbolic $\mathbb{A}\mathbbm{d}\mathbb{S}^3$ manifold homeomorphic to $S\times[-1,1]$, with smooth, space-like and…

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…

Let $\rho: \Gamma\to\operatorname{SO}_{0}(2,n)$ be a GHC-regular representation. A representation is quasi-Fuchsian if it is a deformation of the inclusion of a lattice in…

Let $M=\mathbf{PSL}_2\mathbb{R}/\Gamma$ be a complete Lorentzian manifold, where $\Gamma$ is a cocompact subgroup acting by right translation, and consider deformations whose left…