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. http://resolver.caltech.edu/CaltechETD:etd-06192007-145905

Abstract

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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 [...] norm of the curvature tensor for fixed p > n/2 has a subsequence which converges in [...]. If n = 3, we have the same result if we assume, in addition, that the scalar curvature has an [...] 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