53 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…

Let a gravitational instanton be a complete hyperKähler manifold of ALG or ALG$^*$ type, and let a Hitchin moduli space be a moduli space of solutions to a Hitchin system equipped…

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 a Hermitian gravitational instanton with either ALF or AF asymptotics. Aksteiner–Andersson toricity conjecture. Every such gravitational instanton is toric. The conj…

Hermitianity conjecture. Any ALF instanton must be Hermitian, either Kähler or non-Kähler.

Cherkis–Kapustin classification conjecture. Every ALF-$D_k$ instanton is exactly the metric constructed by Cherkis and Kapustin.

Schwarzschild unit-energy uniqueness conjecture. The positive chirality spin connection is the only unit-energy instanton over the Schwarzschild space.

The ALG gravitational instantons under discussion arise as moduli spaces of periodic monopoles on $\mathbb R^2\times S^1$, while the corresponding moduli spaces of solutions to Hit…

Let $({\cal M},g_{\mu\nu})$ be a space satisfying the geometric and asymptotic conditions specified in the source for the Riemannian Schwarzschild and Taub-bolt instantons. Uniquen…

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…

ALG gravitational-instanton construction conjecture. All ALG gravitational instantons have constructions as moduli spaces of Higgs bundles.

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 hyperkähler gravitational instanton with quadratic volume growth. Modularity conjecture. The gravitational instanton arises as a Hitchin moduli space of Higgs bund…

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 $\beta\in(0,1)$ and $n\geq 1$, and let $X_n$ be the manifold in the source. Let $g_{\beta,n}$ denote the metric constructed in Theorem. AF$\beta$ uniqueness conjecture. The met…

Let $M$ be the total space of $\mathcal{O}(1)$ over $\mathbb{CP}^1$, equipped with an $\operatorname{ALF}$-$A_k$ gravitational-instanton metric $g$. Taub-Bolt uniqueness conjecture…

Let $k\geq 0$, and let $(M,g)$ be an $\operatorname{ALF}$-$A_k$ gravitational instanton. ALF-$A_k$ uniqueness conjecture. Up to isometry, scaling, and orientation reversal, $(M,g)$…

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 $M$ 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…

Hyper-Kähler conjecture. Any Ricci-flat ALE instanton must be locally hyper-Kähler.

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

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…

A Ricci-flat ALE instanton is a complete Riemannian manifold with vanishing Ricci curvature and asymptotically locally Euclidean geometry, approaching $\mathbb{R}^4/\Gamma$ at infi…