Compactness of Conformal Metrics with Integral Bounds on Curvature

Author: Gursky, Matthew J.

Year: 1991

Degree: Dissertation (Ph.D.)

Advisors: Chang, S. Y. A.; Wolff, Thomas H.

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/00WZ-PH51

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 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.

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