6 problems

Let $\Omega$ be a bounded open subset of $\mathbb R^n$ whose boundary $\Sigma=\partial\overline{\Omega}$ is inner embedded, meaning that $\Sigma$ touches itself from the exterior r…

Let $(M,\bar g,\bar u)$ be an asymptotically flat, static vacuum triple, and let the geometric boundary value problem be … where the interior equations are…

Let $\Sigma=\partial\overline{\Omega}$ be an inner embedded hypersurface, meaning that $\Sigma$ touches itself from the exterior region $\mathbb{R}^n\setminus\Omega$. Anderson–Jaur…

Let $(\gamma,H)$ be Bartnik boundary data on $S^2$, normalized by $\operatorname{area}_{\gamma}(S^2)=1$. A maximal static vacuum solution is an asymptotically flat static vacuum tr…

Let $\gamma\in\operatorname{Met}^{m,\alpha}(S^2)$ have non-constant Gauss curvature, and let $C_+^{m-1,\alpha}(S^2)$ denote the positive mean-curvature data. Non-surjectivity of th…

Let $(\gamma,H)$ with $H>0$ be the Bartnik boundary data of a locally flat 3-ball that is not an embedded ball in $\mathbb R^3$. Let ${\mathcal M}$ denote the space of Bartnik boun…