60 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…

Let $M$ be a closed Lorentzian conformal manifold. It is essential if its conformal transformation group does not preserve any metric in the conformal class. Lorentzian Lichnerowic…

CMC existence conjecture. If $(M,g)$ is future timelike geodesically complete and satisfies the strong energy condition, then $(M,g)$ contains a CMC Cauchy surface.

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…

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

Benedetti–Guadagnini conjecture. For every $\gamma\in\pi_1(M)$,

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 $D\subset \mathcal V$ 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 a spacetime be a constant scalar invariant (CSI) spacetime, meaning that all scalar polynomial curvature invariants are constant. A VSI spacetime is one for which all scalar po…

Let a spacetime be a constant scalar invariant (CSI) spacetime, meaning that all scalar polynomial curvature invariants are constant. The higher-dimensional Kundt CSI class consist…

Let a spacetime have a Riemann tensor $R$ and all of its covariant derivatives. A null frame is a frame adapted to the Lorentzian structure, and the boost order and boost weight re…

Let a Lorentzian spacetime $\mathcal{M}$ be a constant scalar invariant (CSI) spacetime, with curvature invariants $\mathcal{I}$. A spacetime is CSI when all scalar polynomial curv…

Let $M$ be a stably causal Lorentzian manifold whose sectional curvatures are much less than $1$, and let $\gamma$ be a timelike curve from $s$ to $t$ in $M$ whose radius of curvat…

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…

Let $(X,\mathsf d,\mathfrak m, \ll, \leq, \tau)$ be a measured Lorentzian space, let $V\subset X$ be the achronal set occurring in the hypotheses of Theorem, and let…

Let $(M,h)$ be an $n$-dimensional Riemannian manifold, and let $g$ be a Lorentzian metric such that $(M,g)$ is a globally hyperbolic spacetime. Write…

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…

Bombelli's conjecture. If, for every finite poset $C$, the probabilities that the samples from $M$ and $N$ are order-isomorphic to $C$ coincide, then $M$ and $N$ are isometric.

Generalized-cone characterization conjecture. The space $Y$ satisfies $\textnormal{\textsf{TCD}}(-\kappa N,N+1)$ if and only if $f”-\kappa f\leq0$ in the distributional sense in…

Let $(M,g)$ be a causal spacetime satisfying the No Observer Horizons condition (NOH). Its conformal group is denoted by $\mathrm{Conf}^\uparrow(M,g)$, and it is essential if there…

Let $(M,g)$ be a pseudo-Riemannian manifold. Its conformal group is denoted by $\mathrm{Conf}^\uparrow(M,g)$, and this group is essential if there is no metric conformal to $g$ for…

Let $X$ be a compact Kähler manifold, let $\Omega$ be a Lorentzian class of dimension $d$ on $X$, and let $\alpha,\beta$ be big nef classes on $\Omega$. Define…

Let $M$ be a globally hyperbolic spacetime with compact Cauchy surfaces satisfying the strong energy condition. Bartnik Splitting Conjecture. If $M$ is timelike geodesically comple…

Lorentzian warped-product timelike curvature-dimension conjecture. If $f”-Kf\leq0$ and $(F,d_F,\operatorname{m}_F)$ is non-branching and satisfies $\mathsf{CD}(K_F(N-1),N)$, then t…