In this thesis, we show that a sequence of conformal metrics on a compact n-dimensional Riemannian manifold (n ≥ 4) which has an upper bound on volume and an upper bound on the LP[...] norm of the curvature tensor for fixed p > n/2 has a subsequence which converges in Cα. If n = 3, we have the same result if we assume, in addition, that the scalar curvature has an L2 bound.
As corollaries, we have the compactness of a sequence of conformal metrics on a compact three-manifold which are isospectral with respect to either the standard or conformal Laplacian, and the result of Lelong-Ferrand that any compact manifold with non-compact conformal group is conformally equivalent to the standard sphere.
