200 problems

Stability conjecture. The family $\mathcal{B}_\Lambda$ is a stable family of solutions to the Einstein vacuum equations with $\Lambda\geq 0$: the evolution of the small perturbatio…

Let $(M^d,g)$ be a complete asymptotically Euclidean manifold of real dimension $d$ with non-negative scalar curvature and an outermost minimal hypersurface $\Sigma$. Write…

Let $\mathcal{S}$ be a generic spacelike initial cosmological singularity, and let $J^+(p)$ denote the causal future of a point $p\in\mathcal{S}$. BKL proposal. Sufficiently close…

Let a gravitational-collapse spacetime arise from one-ended initial data on a hypersurface $\Sigma$. For a point $p$ on a spacelike singularity $\mathcal{S}$, say that its causal p…

Generic transverse blow-up conjecture. For generic data on $\Sigma$, the transverse derivatives of $\phi$ blow up on $\mathcal C\mathcal H^+$; in fact, $\phi$ is not in…

Consider smooth developments of the Einstein–Maxwell–Klein–Gordon system as in Theorem 1.1, with coupling parameter $\mathfrak{e}\neq 0$. Let $\mathcal{Q}^+$ be the quotient spacet…

Uniqueness conjecture for static extensions. $(\Omega,g)$ determines a unique static asymptotically flat manifold $(S,g)$ with boundary $\Sigma\simeq\partial\Omega$ satisfying thes…

Hamiltonian volume conjecture. The rescaled volume satisfies

Let $\mathcal{A}$ be the set of spherically symmetric marginally trapped spheres, let $v$ be the null coordinate used in the paper, and let $V$ be sufficiently large. Achronality c…

Let $(M^3,g,h)$ satisfy $\mu \ge |J|$, and suppose there is a nested sequence of connected legal surfaces … with $\bigcup_iD_i=M$ and $\lim |\Sigma_i|=\infty$. Here legal means out…

Let $(M^3,g,h)$ be asymptotically flat with total ADM mass $m_{ADM}$. Define the total inner mass $m_{inner}^{total}$ and total outer mass $m_{outer}^{total}$ as the suprema of the…

Let $(M^3,g,h)$ be complete, asymptotically flat Cauchy data with $\mu \ge |J|$ and more than one end. Choose one end as special, and let $A$ be the minimum area required to enclos…

A spacetime is maximally extended if it admits no further extension, and it is globally hyperbolic if it has a Cauchy surface. A constant mean curvature (CMC) hypersurface is a spa…

Consider generic initial data for a suitable Einstein-matter system or for the Einstein vacuum equations, and let the maximal Cauchy development be the spacetime arising from those…

Let $(\mathcal{M},g)$ be a $3+1$-dimensional Lorentzian spacetime solving the Einstein field equations, arising as the maximal future development of generic, regular, asymptoticall…

Let $(\gamma,k)$ be the initial data from Theorem, with ADM mass $\mathfrak m_0$ and vanishing linear momentum, let $g$ be the resulting spacetime metric, and set…

Let $(\gamma,k)$ be initial data as in Theorem, let $\mathcal E_0$ be an index set with $\min\operatorname{Re}\mathcal E_0>1$, and suppose … Let $g$ be the spacetime metric produce…

Let $(M^3,g,\mathbf{k})$ be a complete connected asymptotically flat initial data set satisfying the dominant energy condition. Choose an end with ADM mass $m$, and let…

Consider spherically symmetric initial data for the Einstein-scalar field equations in the regularity class $\mathcal{B}$, and let their maximal globally hyperbolic development (MG…

Let $\alpha\geq 0$, and let $(M^3,g)$ be an asymptotically flat manifold with nonnegative scalar curvature containing an outermost minimal surface $\Sigma$ with induced metric…

Nonlinear instability of Schwarzschild–AdS. For any $s\in\mathbb N$, there exists an open and dense set $J^{\mathrm{EVE}}_s\subset(0,+\infty)$ of black-hole masses $M$ such that th…

Kerr stability conjecture. The maximal Cauchy development of any initial data set for the Einstein vacuum equations, sufficiently close to a subextremal Kerr initial data set in a…

Consider generic initial data giving rise to a perturbation of a Kerr black hole governed by the Einstein vacuum equations, and let the perturbed spacetime have a Cauchy horizon. I…

Consider the spherically symmetric Einstein--Maxwell charged scalar field (EMCSF) model and an interpolating family of Cauchy data with critical parameter $p_*$, whose critical dev…

Let $\widehat{\mathfrak M}$ be a moduli space of Cauchy data on $\mathbb R^3$ for a matter-model and symmetry-class combination in which an exact extremal Kerr--Newman event horizo…