3 problems

Let $(M,g)$ be a non-flat $\operatorname{AF}_0$ gravitational instanton. Schwarzschild uniqueness conjecture. The metric $g$ is, up to isometry, the Schwarzschild metric. Evidence…

Let $X$ be a conical $\mathcal{AF}$ 4-manifold with a simply connected end, and let its asymptotic cone be the metric cone at infinity. Petrunin–Tuschmann conjecture. There is no c…

Positive mass conjecture. To any AF gravitational instanton $(M,\mathbf{g})$, one can assign a geometric invariant called the mass $m$, coinciding with the invariant $m$ for toric…