143 problems

Let $(M,g)$ be a globally hyperbolic, timelike geodesically complete spacetime with compact Cauchy surfaces satisfying the strong energy condition. For a Cauchy temporal function…

Lorentzian Lichnerowicz conjecture for conformal vector fields. If $(M,g)$ is not conformally flat, then every conformal vector field on $M$ is inessential.

Let $(\mathbb{R}^4,g)$ be a Ricci-flat pp-wave with metric … where $H\colon\mathbb{R}\times\mathbb{R}^2\to\mathbb{R}$. Ehlers–Kundt conjecture. The metric is geodesically complete…

Moncrief–Isenberg conjecture. Every compact Cauchy horizon in a smooth vacuum spacetime is a Killing horizon.

Let $(X,g)$ be a globally hyperbolic spacetime, and let $\mathcal{N}$ be its space of null geodesics. For each point $p\in X$, write $\operatorname{sky}_p$ for its sky, viewed as a…

Let $(\mathcal M,\textsl{g})$ be a Lorentzian manifold satisfying the hypotheses (H1) and (H3), and consider the unique continuation theorem in the exterior of the double null cone…

Let a CSI spacetime be a spacetime all of whose scalar curvature invariants are constant, and let a degenerate Kundt spacetime be a Kundt spacetime whose curvature tensor and all c…

A spacetime with a compact Cauchy horizon is a spacetime admitting a non-empty compact Cauchy horizon. A Killing horizon is a null hypersurface whose generators are tangent to a Ki…

Galloway–Ling conjecture. Then $M$ contains a CMC Cauchy surface.

Let $(M_j,g_j)$ be future Cauchy developments with time functions $\tau_j$, and let $(M_{\infty},g_{\infty})$ be a spacetime with time function $\tau_{\infty}$. For $T$ in the rele…

Let $(M,g)$ be an asymptotically static spacetime and let $\square_g$ denote its wave operator. Essential self-adjointness conjecture. The operator $\square_g$ is essentially self-…

Let a Lorentzian conformal manifold be a manifold equipped with a conformal class of Lorentzian metrics, and let $\operatorname{Ein}^{1,n}$ denote the Lorentzian Einstein space. Tw…

A spacetime is a pair $(M,g)$, where $M$ is a differentiable manifold and $g$ is a smooth, symmetric, non-degenerate $(0,2)$-tensor field. Two spacetimes are isocausal when there a…

Let a Morse spacetime be a singular Lorentzian spacetime built from a Morse function, whose critical points have an index and a coindex. Borde–Sorkin conjecture. A Morse spacetime…

Zeghib's conjecture. All compact quotients are standard.

Let $(M,g)$ be a causally simple spacetime, and let $\mathcal{N}_g$ denote its space of null geodesics. A conformal embedding is an embedding of $(M,g)$ into another spacetime that…

In de Sitter space, let $M$ be a complete spacelike hypersurface with constant mean curvature. Goddard's conjecture. $M$ must be totally umbilic. This is a conjecture in Lorentzian…

An $\mathcal{I}$-non-degenerate spacetime is one for which the scalar polynomial curvature invariants characterize the spacetime locally up to the relevant equivalence. Cartan-inva…

Conjecture on three-quarters supersymmetry. There exist backgrounds with $\nu=\tfrac34$ which are not locally homogeneous.

Let $(M,\mathcal{C})$ be a Riemann-Lorentz conformal space with polar end $D^{\infty}$. Suppose that $D^{\infty}$ is conformal-flat. A simultaneity distribution is a distribution w…

Let $(\Sigma,h)$ be a connected Riemannian $3$-manifold and let $\sigma$ be a symmetric two-covariant tensor satisfying the compatibility conditions of a second fundamental form. L…

Let $D\subset \mathcal V$ be a simply connected domain of an $n$-dimensional $k$-recurrent Lorentzian manifold $(\mathcal V,g)$. A vector field is null recurrent if it is null and…

Let $D\subset \mathcal V$ be a simply connected domain of an $n$-dimensional $k$-symmetric Lorentzian manifold $(\mathcal V,g)$. A vector field is null covariantly constant if it i…

Let $\mathcal{M}:=(M,g)$ be a pseudo-Riemannian manifold of Lorentzian signature $(1,q)$. For $P\in M$, let $g_P:=g|_{T_PM}$ and let $\nabla^iR_P:=\nabla^iR|_{T_PM}$. Its $k$-model…

The esc conjecture. Every connected component of the space of Lorentzian metrics on $M$ contains a metric with scalar curvature $s$.