A Gap Theorem For Half-Conformally-Flat 4-Manifolds

Given a smooth, compact manifold, an important question to ask is, what are the best'' metrics that it admits. A reasonable approach is to consider as best'' metrics those that have the least amount of curvature possible. This leads to the study of canonical metrics, that are defined as minimizers of several scale-invariant Riemannian functionals. In this dissertation, we study the minimizers of the Weyl curvature functional in dimension four, which are precisely half-conformally-flat metrics. Extending a result of LeBrun, we show an obstruction to the existence of ``almost'' scalar-flat half-conformally-flat metrics in terms of the positive-definite part of its intersection form. On a related note, we prove a removable singularity result for Hodge-harmonic self-dual 2-forms on compact, anti-self-dual Riemannian orbifolds with non-negative scalar curvature.