22 problems

In the setting of Theorem, let $\bm{l}=\bm{l}(\beta)$ be the length vector associated with the construction of $\operatorname{AF_\beta}$ gravitational instantons. Uniqueness conjec…

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 a gravitational instanton whose underlying manifold $M$ is topologically $\mathbb{R}^4$. Topological $\mathbb{R}^4$ rigidity conjecture. The metric $g$ is either the…

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 $(M,g)$ 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 $(M,g)$ be a gravitational instanton. Finite-topological-type conjecture. The manifold $M$ has finite topological type, meaning that it can be compactified into a smooth manifo…

Let $(M,g)$ be a gravitational instanton. Asymptotic-cone uniqueness conjecture. The gravitational instanton $(M,g)$ has a unique asymptotic cone. Uniqueness is known for complete…

Let $(M,g)$ be a gravitational instanton, and let an asymptotic cone mean a pointed-Gromov–Hausdorff limit obtained by rescaling $(M,g)$ along a sequence tending to infinity. Metri…

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…

Let $p_i$ be punctures with parabolic weights $\boldsymbol{\alpha}=(\alpha_i)$, and call the flags not all full when, for some $p_i$, the corresponding weight satisfies…

A simply-connected Ricci-flat ALE instanton is a complete Riemannian manifold with vanishing Ricci curvature and asymptotically locally Euclidean geometry. Gibbons–Bando-Kasue-Naka…

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…

ALF Hermitianity conjecture. Every ALF gravitational instanton is Hermitian.

Killing-field classification conjecture. The conclusion of the Hermitian classification conjecture holds for $(\mathcal{M},g_{ab})$.

Hermitian classification conjecture. $(\mathcal{M},g_{ab})$ is one of Kerr, Chen--Teo, Taub-bolt, or Taub--NUT with the orientation opposite to the hyperkähler orientation.

Consider a sequence of pointed semi-flat $ALH_d^*$-gravitational instantons satisfying the assumption of Theorem, with parameters $c_i$ and $\tau_i$. Collapsing limits conjecture.…

An $ALG$ or $ALG^*$ gravitational instanton may, after a suitable hyperKähler rotation, admit a compactification obtained by adding a canonical cycle to a rational surface. Compact…

Let $(\mathcal{M}, g_{ab})$ be a Hermitian, non-Kähler, ALF gravitational instanton, meaning a complete Ricci-flat Riemannian four-manifold with the stated Hermitian, non-Kähler, a…

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…

Let $e$ be an exotic $\mathbb{R}^{4}$ belonging to the DeMichelis–Freedman radial family, and let the foliated topological limit of general relativity (FTL of GR) on $e$ be evaluat…