Moduli spaces of critical Riemannian metrics with norm curvature bounds

We consider the moduli space of a class of critical metrics on compact manifolds that includes extremal Kähler metrics. We show that under the conditions of volume and diameter bounds, curvature bounds, and Sobolev constant bounds, this Moduli space can be compactified by including orbifolds with finitely many orbifold points. In dimension >4 these orbifold points are C∞ Riemannian orbifold points, and in the 4 dimensional extremal Kähler case, they are also of class C∞.