Compactness of Conformal Metrics with Integral Bounds on Curvature

Citation

Gursky, Matthew J. (1991) Compactness of Conformal Metrics with Integral Bounds on Curvature. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/00WZ-PH51. https://resolver.caltech.edu/CaltechETD:etd-06192007-145905

Abstract

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 L^P[...] norm of the curvature tensor for fixed p > n/2 has a subsequence which converges in C^α. 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.

Thesis Files