Many geometric problems require the passage from very natural energy bounds on curvature quantities, such as L2 control on the Riemann tensor, to very unnatural pointwise bounds. The long-standing approach has been the use of analysis, yet this relies on the persistence of a geometric-analytic nexus, the Sobolev constant, that is difficult or impossible to control in most situations. In this talk we discuss a more intrinsically geometric way of obtaining a priori regularity of critical 4-manifolds that circumvents use of the Sobolev constant. Some background and uses of Riemannian moduli spaces will be discussed, particularly questions of Kahler geometry that have received a lot of attention recently.