Einstein Metrics and Metrics with Bounds on Ricci Curvature

Michael T. Anderson
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Michael T. Anderson: State University of New York, Department of Mathematics

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 443-452 from Springer

Abstract: Abstract There is a well-developed theory of the behavior of Riemannian metrics on smooth manifolds, which have uniform bounds on the sectional curvature K. The compactness theorem of Cheeger-Gromov [Ch], [Gr] implies that the space of metrics satisfying the bounds 0.1 $$\left| K \right| \leqslant \Lambda ,{\text{vol}} \geqslant v,{\text{diam}} \leqslant D$$ | K | ≤ Λ , vol ≥ v , diam ≤ D is C1, α precompact. Thus, given any sequence of metrics g i satisfying the bounds (0.1), there is a subsequence {i′} and a sequence of diffeomorphisms φL′ of M, such that the isometric metrics g′ p = (φi′)*gi′ converge, in the C1, α′ topology on M, to a C1, α metric g∞ on M, ∀α′

Date: 1995
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DOI: 10.1007/978-3-0348-9078-6_37

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