| A gravitational instanton is a complete, noncompact, Ricci-flat, Kahler manifold of real dimension four whose curvature tensor has finite L2 norm. Such spaces would seem to play a key role in the structure theory of Riemannian 4-manifolds with bounded Ricci curvature, but their classification has remained largely elusive except for the asymptotically locally Euclidean (ALE) case. It has been conjectured that there exist four main families (ALE, ALF, ALG, ALH) supposedly characterized by having local Rk x T4--k asymptotics. However, examples with k ∈ {1, 2} are scarce.;(1) We construct new families of ALG and ALH gravitational instantons by removing fibers from rational elliptic surfaces, and we examine their geometry. We expect these families to be exhaustive up to specific deformations. The construction also yields a somewhat unexpected class of examples with zero injectivity radius, volume growth rate 43 , and precisely quadratic curvature decay.;(2) In the course of this construction, we obtain sharp weighted Sobolev inequalities on a fairly general class of complete Riemannian manifolds of polynomial volume growth. We apply these to prove a precise existence result for bounded solutions to the complex Monge-Amp`ere equation, thus improving work due to Tian-Yau, and also to obtain reasonable decay rates for these solutions.;(3) We show that a Riemannian metric with Scal ≥ 0 on R4 which converges to the Taub-NUT ALF gravitational instanton at a rate of r--1--delta and is uniformly sufficiently close to Taub-NUT must already be isometric to Taub-NUT. A stronger result with a much simpler proof was independently obtained by Vincent Minerbe [86]. |
