A gap theorem for half-conformally flat manifolds

Abstract

We show that a compact half-conformally flat manifold of negative type with bounded $L^2$ energy, sufficiently small scalar curvature, and a noncollapsing assumption has all Betti numbers bounded in terms of the $L^2$ curvature norm. We give examples of multi-ended bubbles that disrupt attempts to improve these Betti number bounds. We show that bounded self-dual solutions of $d\mathit{\omega }=0$ on asymptotically locally Euclidian (ALE) manifold ends display a rate-of-decay gap: they are either asymptotically Kähler, or they have a decay rate of $O(r^{-4})$ or better.