48 problems

Penrose conjecture. If $\Sigma$ is a past apparent horizon, then

EMD charged Riemannian Penrose inequality. If the charge densities $\operatorname{div}(e^{-2\phi}E)$ and $\operatorname{div}B$ are compactly supported, then

An asymptotically flat support surface is a surface $S\subset\mathbb{R}^3$ with the asymptotic and curvature properties described in Volkmann's theory. The Riemannian Penrose inequ…

Let $(M^{3},\gamma)$ be an asymptotically hyperbolic manifold whose boundary $\partial M$ consists of an outer-minimizing horizon. Denote by $m_{\textnormal{AH}}(M^{3},\gamma)$ its…

Let $(M,g,E,0)$ be a strongly asymptotically flat, time-symmetric initial data set for the Einstein–Maxwell equations, with horizon components indexed by $i$. Let $m$ be the mass,…

Let $(M,g,E,0)$ be a strongly asymptotically flat time-symmetric initial data set for the Einstein-Maxwell equations, with mass $m$ and horizon components having area radii $R_i$ a…

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…

Let $(M^3,g,h)$ be complete, asymptotically flat Cauchy data with $\mu \ge |J|$, and let $\Sigma$ be an apparent horizon satisfying $H_\Sigma=\operatorname{tr}_\Sigma(h)$. Let $m$…

Let $(M^3,g,h)$ be asymptotically flat Cauchy data satisfying $\mu \ge |J|$, and suppose the Einstein evolution from these data has the global properties described in the Penrose a…

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…

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…

Universal black-hole inequality conjecture. The mass should satisfy

Let $M$ be the mass, $A$ the horizon area, $\alpha$ the Einstein–Gauss–Bonnet coupling, and $\chi(\Sigma)$ the Euler characteristic of the horizon. Gauss–Bonnet Penrose conjecture.…

Let $S=A/(4G\hbar)$ be the Bekenstein–Hawking entropy of a black hole with mass $M$ and horizon area $A$. Entropy–mass bound conjecture. The entropy should satisfy … with equality…

Let $\Sigma$ be a compact surface in an asymptotically flat manifold with nonnegative scalar curvature $R\geq0$. Let $M$ be the mass and define the Schwarzschild radius by $r_H=2M$…

Generalized charged Penrose conjecture. The ADM mass should satisfy

Let the initial data contain $n$ disjoint outermost marginally outer trapped surfaces (MOTS) $\{\Sigma_i\}_{i=1}^n$, with areas $A_i$ and angular momenta $J_i$. Multi-horizon exten…

Consider initial data satisfying the appropriate energy conditions, with horizon area $A$, angular momentum $J$, and electric charge $Q$. Kerr–Newman extension conjecture. The ADM…

Let $M_{irr}=\sqrt{A/(16\pi)}$ be the irreducible mass, let $J$ be angular momentum, let $Q$ be electric charge, and let $E_{EM}$ denote the electromagnetic contribution. Kerr–Newm…

Let the initial data be axisymmetric and satisfy the dominant energy condition, but not necessarily the vacuum condition, and let $\Sigma$ be an outermost stable marginally outer t…

Let $(M,g,K) fin \mathcal{M}_{\textnormal{AF}}$ and let $\{\Sigma_r\}_{r\in[r_0,\infty)}\subset\mathcal{M}$, where … Suppose that the inner boundary $\Sigma_{r_0}$ is a generalized…

Let $(M^n,g)$ be a complete, asymptotically flat manifold with non-negative scalar curvature and non-empty minimal boundary $\Sigma$, where $3\leq n\leq 7$, and assume … Here $C_p$…

Let $(M,g,k)$ be a spherically symmetric initial data set, and consider solutions $(f,\phi)$ of the coupled system … … with the appropriate asymptotics for application to the Penro…

Riemannian Penrose inequality without horizon. One should have

Let $(M_i,g_i)$ be a sequence of asymptotically flat $3$-manifolds with nonnegative scalar curvature and ADM mass $m(g_i)$ tending to $0$. Suppose $Z_i\subset M_i$ are the subsets…