6 problems

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

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 $\Sigma$ 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…

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…

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 $F^*(g_{\mathrm{Eucl}})$.…