7 problems

Let $(M,g)$ be a complete oriented Ricci-flat 4-manifold. Finite-topological-type characterization. The manifold $(M,g)$ is a gravitational instanton if and only if $M$ has finite…

Let $(M,g)$ be an ALE gravitational instanton. Bando–Kasue–Nakajima conjecture. The gravitational instanton $(M,g)$ is of Type I, up to orientation reversal. Partial results are kn…

Let $M$ be a complete Ricci-flat Kähler manifold. Yau's compactification conjecture. $M$ can be compactified into a compact complex manifold by adding a divisor at infinity. This c…

Type II classification conjecture. All Type II gravitational instantons are toric and must be ALF or AF, except the Eguchi-Hanson metric with the reversed orientation.

Finite-energy versus finite-topology conjecture. The manifold $(X,g)$ has finite energy if and only if $X$ has finite topological type.

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…

Non-existence conjecture. There exist no Hermitian non-Kähler ALE gravitational instantons, without any restrictions on the structure group, except for the Eguchi–Hanson metric wit…