8 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…

Flat and Schwarzschild uniqueness conjecture. The standard flat exterior boundary data $(\gamma_{+1},n)$, and likewise the standard Schwarzschild exterior boundary data…

For $m\geq 0$, let … with static potential … and consider the exterior region $M=r^{-1}[1,\infty)$. Its boundary data are $(\gamma,H)=(g_{S^n(1)},n\sqrt{1-2m})$. A point of the Bar…

Let $g_{Sch}(m)$ be a Schwarzschild metric with radius parameter $R=1$, and let $\Pi$ be the boundary-data map. A metric is a regular point of $\Pi$ when the differential of $\Pi$…

Let $N=D^2\times S^2$ with horizon $\mathcal H=S^2$. Let $\gamma_{+1}$ be the unit round boundary metric, let $[c]$ denote the relevant boundary conformal or scaling datum, let…

Flat boundary-data extension conjecture. Proposition 4.3 should hold for any such flat boundary data $(\gamma,H)$. The conjecture proposes extending the proposition from the explic…

Let $M=B^3$ and let $N=S^2\times D^2$ with boundary $\partial N=S^2\times S^1$, in the Schwarzschild black hole case. Let $\Pi$ be the boundary-data map from static vacuum metrics…