Brian J. WeberCitoler-Saumell, Martin2023-05-222001-01-012018-02-232016-01-012018-02-23https://repository.upenn.edu/handle/20.500.14332/29129Given 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.94 p.application/pdfMartin Citoler-Saumellanti self dualepsilon regularityhalf conformally flatHodge harmonicorbifoldsignatureMathematicsDissertation/Thesis