30 problems

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 $(M^3,g,\mathbf{k})$ 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 $(M,g,K) fin \mathcal{M}_{\textnormal{AF}}$ 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…

Let $(M,g,k)$ be an asymptotically flat initial data set satisfying appropriate fall-off conditions and the dominant energy condition, with boundary $\partial M$ consisting of an o…

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…

Let $M$ be the ADM mass, $\kappa$ the surface gravity, and $\mathcal{A}$ the event-horizon area of a 4-dimensional asymptotically flat, static or axisymmetric stationary black hole…

Let $(M,\gamma,E,\Lambda)$ be an electrically charged Riemannian manifold of dimension $n+1\geq3$ which is geodesically complete up to its closed, strictly outward minimizing minim…

Let $S$ be an asymptotically flat support surface with mean curvature $H^S\geq 0$. Let $S'\subset S$ be an exterior surface with free boundary minimal surface $\Sigma$ and exterior…

Let $n$ and $q$ be integers with $n\geq 3$ and $1\leq q<n/2$. Let $(M^n,g)$ be asymptotically flat of order $\tau>(n-2q)/(q+1)$, with integrable nonnegative $q$-th Gauss–Bonnet cur…

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

Let $(M^n,g)$, $n \ge 3$, be a complete, asymptotically flat, smooth $n$-manifold with nonempty compact boundary $\Sigma=\partial M$ smooth with respect to both the smooth structur…

Bray–Iga conjecture. The Riemannian Penrose inequality holds for arbitrary-dimensional conformally flat, asymptotically flat manifolds with nonnegative scalar curvature.

Mixed Riemannian ZAS inequality conjecture. The ADM mass satisfies

Let $(\Sigma,\gamma_{ij},A_{ij})$ be an asymptotically Euclidean initial data set that is Schwarzschild at infinity, with total mass $M_{ADM}$ in a chosen end, and satisfying the d…

Let $(M,g,k)$ be asymptotically flat Cauchy data with an outermost generalized apparent horizon boundary $\Sigma$. Define … … where $v=\phi\nabla f/(1+\phi^2|df|_g^2)^{1/2}$. Jang–…