7 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…

ALE hyperkähler conjecture. Any ALE Ricci-flat $4$-manifold must be hyperkähler.

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

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…

Stable ALE desingularization conjecture. There exists an ancient, respectively immortal, renormalized Ricci flow on $M_o\#_{\mathbb{S}^3/\Gamma}N$ with limit $(M_o,g_o)$ at…

A Ricci-flat metric is a metric with vanishing Ricci curvature; the asymptotic model is the quotient cone $\mathbb{R}^4/\mathbb{Z}_2$, and the Eguchi–Hanson metric is a Ricci-flat…