8 problems

Let $\Omega$ be a compact Riemannian $3$-manifold with boundary, and consider asymptotically flat extensions satisfying the natural Bartnik boundary constraints. The infimum of the…

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

Vacuum constraint extension conjecture. If $(M,g)$ is an extension of $(\gamma,H)$ in $\mathcal P$, then there exists an extension in $\mathcal C_{(\gamma,H)}$; equivalently,

Let $\Omega$ be a compact domain with boundary data induced by a nonnegative-scalar-curvature metric, and let … where $\mathcal C_{(\gamma,H)}$ is the set of asymptotically flat ex…

Let $\Sigma$ be a surface, and let its full (spacetime) Bartnik mass be defined using admissible asymptotically flat initial-data extensions with the prescribed boundary data. A st…

Let $(\Omega,\gamma)$ be a compact Riemannian manifold with boundary $\Sigma$. Let $g$ be the metric induced by $\gamma$ on $\Sigma$, and let $H$ be the outward mean curvature of…

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 $F^*(g_{\mathrm{Eucl}})$.…

Let $\Omega_0$ be a region and let $\mathcal{PM}(\Omega_0)$ denote the class of asymptotically flat extensions containing an isometric image of $\Omega_0$. Bartnik extension compac…