4 problems

Cosmic no-hair conjecture. The only $n$-dimensional compact static triple $(M^n,g,f)$ with positive scalar curvature and connected boundary $\Sigma$ is a round hemisphere…

Uniqueness conjecture for static extensions. $(\Omega,g)$ determines a unique static asymptotically flat manifold $(S,g)$ with boundary $\Sigma\simeq\partial\Omega$ satisfying thes…

Let $(M^n,g,f)$ be an $n$-dimensional compact static triple with positive scalar curvature and connected boundary $\Sigma$. A static triple is a complete and connected Riemannian m…

Let $(M^n,g,f)$ be a positive static triple: $M$ is a connected smooth $n$-manifold with boundary, $g$ is a complete Riemannian metric, and $f\in C^\infty(M)$ is a nontrivial non-n…