We show that a compact half-conformally flat manifold of negative type with bounded energy, sufficiently small scalar curvature, and a noncollapsing assumption has all Betti numbers bounded in terms of the 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 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 or better.
"A gap theorem for half-conformally flat manifolds." Illinois J. Math. 65 (1) 71 - 96, April 2021. https://doi.org/10.1215/00192082-8886951