14 problems

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…

A four-dimensional gravitational instanton is a complete, oriented Riemannian Einstein manifold. In the Ricci-flat case, an instanton is asymptotically flat (AF) if it has the stan…

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 $M$ 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…

Kerr uniqueness conjecture. The Riemannian version of the Kerr solution should be the unique AF instanton.

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…

A Ricci-flat ALE metric is a Ricci-flat metric on an asymptotically locally Euclidean space. Kählerness conjecture. The only Ricci-flat ALE metrics are Kähler, and hence integrable…

A Ricci-flat ALE space is a complete Ricci-flat manifold that is asymptotically locally Euclidean; the paper refers to gravitational instantons and their Kähler quotients as the tw…

Let $G1$ through $G6$ denote the compact, orientable flat three-manifolds described in the source, and consider their spin structures; among these manifolds, $T^3$ is the three-dim…

A Ricci-flat manifold is non-supersymmetric when it does not admit the parallel spinor structure required for supersymmetry, and a manifold is unstable when it admits a non-perturb…

A compact Ricci-flat manifold is a compact manifold with a Ricci-flat Riemannian metric, and it is stable when it has no relevant instability; a Riemannian manifold has special hol…