17 problems

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…

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

Monotonicity conjecture. The quasi-local mass $m(\Sigma)$ has increasing monotonicity.

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 $\Sigma_s$ be a family of surfaces, let $m_{WY}(\Sigma_s)$ denote the Wang–Yau quasi-local mass, and let $E(\Sigma_s,\tau)$ denote the Wang–Yau quasi-local energy for an admiss…

Let $\Sigma_s$ be a family of surfaces approaching an apparent horizon, and let the optimal embedding equation be the Euler–Lagrange equation for the Wang–Yau quasi-local energy. L…

Let a family of surfaces approach an apparent horizon in a spacetime, and let the optimal embedding of each surface be its embedding into the reference Minkowski space that solves…

Let $(M,g)$ be a manifold with boundary $\partial M=S\cup\Sigma$, where $S$ and $\Sigma$ are smooth discs meeting orthogonally along $\partial\Sigma=\partial S$. Assume that…

Strict Bartnik mass bound conjecture. Under these hypotheses, strict inequality should hold:

Let $\Sigma$ be a $2$-surface bounding a finite domain in a $3$-manifold $(M,g)$ of nonnegative scalar curvature. Let $(\mathbb{N},\bar g)$ be a complete, asymptotically flat $3$-m…

Let $F:\bar B\rightarrow\mathbb R^3$ be a smooth immersion of a 3-ball that is not an embedding, and equip the ball with the pulled-back Euclidean metric $F^*(g_{\mathrm{Eucl}})$.…

In general relativity, consider a finitely extended spacetime region with boundary. A quasi-local mass and other conserved quantities are expected to be associated with the region…

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 $(\Sigma,\gamma,H)$ be Bartnik data, and define … where … A nonnegative scalar curvature fill-in is a compact Riemannian $3$-manifold with boundary whose scalar curvature is no…

Let $M$ be a Riemannian $n$-manifold, let $\Omega\subset M$ be path connected, precompact, and open, and let $\Sigma=\partial\Omega$ carry Bartnik boundary data…

Let $(\Sigma,\gamma,H)$ be the boundary data considered in the paper, let $m(\lambda)$ denote the Bartnik inner mass associated with $(\Sigma,\gamma,\lambda H)$, and let…