10 problems

Let $(N,g',y')$ be a blow-up limit given by Theorem 5.1, with maximal domain $N^{o}$ of positivity for the static potential $\bar{u}$, boundary $\partial N^{o}$, and event-horizon…

Consider a perturbed curve $\widetilde L$ in the boundary-data space, transverse to the boundary-data map $\Pi_B$, with connected inverse image $\sigma=\Pi_B^{-1}(\widetilde L)$. S…

Bartnik's diffeomorphism conjecture. The map $\Pi_B$ should be a diffeomorphism. This is presented as a refinement of the conjecture that $\Pi_B$ is a bijection, and would give glo…

Let $n>1$ and consider an asymptotically flat static metric with $n$ boundary components satisfying some separation condition on the boundary components. Static $n$-body conjecture…

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 $\operatorname{Imm}_{A}(S^2,\mathbb R^3)$ be the space of smooth Alexandrov immersed $2$-spheres, and let…

Let $(\gamma,H)$ be data on $\partial\Omega$ with positive Gauss curvature $K_\gamma>0$ and $H>0$. Let $\lambda_0>0$ be the threshold such that the scaled data have a fill-in with…

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…