206 problems

For every admissible Bartnik boundary datum $(\Sigma,\gamma,H)$, the Bartnik mass…

For every admissible compact initial-data set $(\Omega,g,k)$ with boundary $\Sigma$ for which the Wang--Yau quasi-local mass $\mathcal{M}_{\mathrm{WY}}(\Sigma)$ is defined and the…

Conjecture: Every asymptotically flat, stationary, axisymmetric solution $(\mathcal{M},g,F)$ of the Einstein--Maxwell equations representing a regular electrovacuum black hole is u…

Every regular, stationary, asymptotically flat, vacuum black-hole spacetime in four-dimensional general relativity is isometric to a member of the Kerr family, parametrized by mass…

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For every black-hole spacetime with Hawking temperature $T_H$, the least-damped quasinormal mode with frequency $\omega_0$ satisfies…

Positive mass conjecture. The ADM mass of $(M,g)$ is nonnegative. Moreover, if the mass is zero, then $(M,g)$ is isometric to Euclidean space $(\mathbb{R}^n,\delta)$.

Let asymptotically flat initial data for Einstein's field equations be given, and let its maximal development be the corresponding spacetime. A future null infinity $\mathcal{I}^+$…

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 $g$) be a smooth Riemannian metric on the closed hemisphere $\mathbb{S}^3_+$ satisfying the following conditions: its scalar curvature obeys ${\mathrm R}_g\geq 6$; its induced…

Let $(\Omega, g_0, k_0)$ be a compact initial data set with nonempty smooth boundary, satisfying the dominant energy condition, and let $(M, g, k)$ be a Bartnik mass minimizer for…

Gauge-invariant decomposition conjecture. There exist a tensor field ${\cal H}_{ab}$ and a vector field $X^a$ such that

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

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…

Let ${{\mathscr M}}_{\epsilon}$ be a physical spacetime, let ${{\mathscr M}}$ be its background spacetime, and let $h_{ab}$ be a perturbative metric tensor pulled back from…

Burnett's backward conjecture. Every sufficiently regular solution to the Einstein--massless Vlasov system arises as a high-frequency limit of vacuum spacetimes.

Let $\Sigma$ be a two-surface with area $A$ and mean curvature $H$, and define its Hawking mass by … Suppose that $\Sigma$ is a stable marginally outer trapped surface (MOTS) with…

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 $(M,g)$ be the maximal future development of generic asymptotically flat initial data. A terminal indecomposable past set is a past set $\mathcal{P}$ in $M$ of the type describ…

Let $\mathcal{I}^+$ denote future null infinity, and consider the admissible data in the one-ended spherically symmetric Einstein–Maxwell–Klein–Gordon model. Weak Cosmic Censorship…

Let vacuum Cauchy data be sufficiently near data corresponding to a subextremal Kerr metric $g_{a_0,M_0}$ with $|a_0|<M_0$, and let $(\mathcal{M},g)$ be the associated maximal vacu…

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 $\mathcal{S}$ 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…