19 problems

Horowitz–Myers conjecture. Roughly speaking, any complete initial data set for the Einstein equations with flat toroidal conformal infinity should have mass bounded below by the ma…

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 $B^4$ be the $4$-ball, or more generally let $B^{n+1}$ be the $(n+1)$-ball, equipped with the setting of asymptotically hyperbolic Einstein metrics discussed above. Ball cusp-f…

Nonexistence conjecture. No such strictly globally static asymptotically hyperbolic Einstein solutions exist.

Let $(M,g_\alpha)$ be a two-dimensional smooth manifold with a family of gas giant metrics $g_\alpha$. Assume that the function $\beta^{(1)}(\theta)$ from the boundary value proble…

Positive mass conjecture. The mass of an asymptotically hyperbolic vacuum initial data set is positive unless it is a Cauchy hypersurface of Minkowski space.

Let $\rho$ be the energy density and $J$ the momentum density on a smooth, conformally compact, geometrically finite general relativistic initial data set satisfying the energy con…

Let $(M,g)$ be an $n$-dimensional conformally compact ALH manifold without boundary, with well-defined mass $m$ and scalar curvature satisfying … Let the metric at infinity determi…

Let $(X,g)$ be an $n$-dimensional asymptotically hyperbolic manifold with scalar curvature $R_g\geq -n(n-1)$. Its mass is a vector $(p_0,p_1,\dots,p_n)$, and the positive mass theo…

Let $(M,g,\Sigma)$ be an asymptotically hyperbolic manifold with scalar curvature $R_g$ and boundary mean curvature $H_g$ satisfying … For any admissible chart $F$, let…

Riemannian Horowitz-Myers conjecture. The Wang mass-energy $m$ of $(M,g)$ satisfies $m\ge m_0$, where $m_0$ is the mass of the Horowitz-Myers geon having least mass among all Horow…

Let $q$ and $\tilde q$ be potentials for the non-selfadjoint radial equation considered in the paper, with $\tilde q-C(\lambda^2+\frac{1}{4})$ satisfying the isospectrality conditi…

Let $(\Omega,g)$ be a compact $3$-manifold whose boundary $\Sigma$ is topologically a sphere with Gauss curvature $K>-1$ everywhere. Let $\Sigma$ be isometrically embedded in hyper…

Let $(M,g)$ be an asymptotically hyperbolic manifold with scalar curvature $R\geq -6$ and an outermost minimal boundary, and let $m$ be its total mass. The Schwarzschild anti-de Si…

Positive mass conjecture. The vector $\mathfrak m_{\Psi}$ is time-like and future-directed, or it vanishes; in the latter case $(M,g)$ is isometric to $(\mathbb H^n,g_0)$.

Let $M$ be an asymptotically hyperbolic manifold of dimension $n$ whose scalar curvature is at least $-n(n-1)$, and let its mass be defined in the asymptotically hyperbolic setting…

Let $(M,g)$ be a complete, $n$-dimensional Riemannian asymptotically hyperbolic manifold whose conformal infinity is the $(n-1)$-unit sphere. Let $R_g$ denote the scalar curvature…

Let $(M,g)$ be an asymptotically hyperbolic manifold with energy-momentum vector, and let $n$ be its dimension. Positive energy-momentum conjecture. If $(M,g)$ is complete and has…

Let $(M,g)$ be an asymptotically hyperbolic $3$-manifold with scalar curvature $R\geq -6$ and mass $M$. Let $\Sigma_0$ be an outermost sphere with mean curvature $H(\Sigma_0)=2$. W…