4 problems

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 $\gamma\in\operatorname{Met}^{m,\alpha}(S^2)$ 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…