Compactness of conformal metrics with integral bounds on curvature (Q1320594)

The main problem considered in the paper can be roughly stated as follows. Suppose that \((M,g_ 0)\) is a compact Riemannian manifold and that we conformally deform \(g_ 0\) to obtain a new metric \(g\), i.e. that \(g = e^ f g_ 0\) where \(f\) is a smooth function. Suppose further that we have a priori information about \(g\) (e.g., bounds on curvature and volume); can we recover some quantitative information about the conformal factor \(f\)? The author provides a compactness criterion for metrics of a fixed conformal class, assuming only a volume bound and an \(L^ p\)-bound on the curvature.