7 problems

Consider the setting of Theorem … , the spacetime is future causally geodesically complete. The surrounding discussion says that the answer is expected to be affirmative for…

Critical-index conjecture. All solutions exist for all time if $k\geq \frac{n}{2}+1$, whereas for $k<\frac{n}{2}+1$ there are always some solutions that break down in finite time.…

Let $(M,g)$ be a Lorentzian manifold with $M=\mathbb{R}\times\Sigma$ and metric tensor … For each $t\in\mathbb{R}$, let $(\Sigma_t,g_{\Sigma_t})$ denote the Riemannian manifold ind…

Romero–Sánchez conjecture. Null completeness of a compact Lorentzian manifold should imply completeness.

Let $G^1$ and $G^2$ be smooth, weak Riemannian metrics on an infinte-dimensional Banach manifold $\mathcal H$. Assume that $G^2$ is uniformly stronger than $G^1$, both metrics admi…

Geroch's conjecture. Geodesically complete solutions to the Einstein field equations should be rare, in some suitable sense.

RMG completeness conjecture. If $M$ is RMG complete, then it is geodesically complete.