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Zbl 0504.53034
Hamilton, Richard S.
Three-manifolds with positive Ricci curvature. (English)
[J] J. Differ. Geom. 17, 255-306 (1982). ISSN 0022-040X

In this paper the following important theorem is proved: Let $X$ be a compact 3-manifold which admits a Riemannian metric with strictly positive Ricci curvature. Then $X$ also admits a metric of constant positive curvature. As all manifolds of constant curvature have been completely classified by {\it J. A. Wolf} in his book [Spaces of constant curvature. New York etc.: McGraw-Hill (1967; Zbl 0162.53304)], by the preceding theorem these are the only compact three-manifolds, which can carry metrics of strictly positive Ricci curvature. Accordingly, a conjecture of {\it J.-P. Bourguignon} [Global differential geometry and global analysis, Proc. Colloq., Berlin 1979, Lect. Notes Math. 838, 249--250 (1981; Zbl 0437.53011)] is answered affirmatively. As pointed out by the author, this theorem is closely related to the famous Poincaré's conjecture on the compact, simply-connected 3-manifolds and the Smith's conjecture related to the group of covering transformations. If both conjectures were true, the main theorem mentioned above would follow as a corollary. In the proof of this theorem the author studied in advance the equation of evolution $\frac{\partial g_{ij}}{\partial t}=\frac 2n rg_{ij} - 2R_{ij}$, where $r$ is the average of the scalar curvature $R$, namely $r=\int R\,d\mu/\int\,d\mu$. He proved that if for a compact 3-manifold the initial metric has strictly positive Ricci curvature, then it continues like that for ever, and converges to a metric of constant positive curvature, as $t$ tends to $\infty$. Then he made use of the fact peculiar to three dimensions, that the full Riemannian curvature tensor $R_{hijk}$ can be calculated from the Ricci tensor $R_{ij}$ (Weyl's conformal curvature tensor is vanishing). In view of this crucial step, the method used here cannot be generalized to the case $\dim(M)\geq 3$, unless it is extensively modified.

Display scanned Zentralblatt-MATH page with this review.
[C.-c. Hwang]

MSC 2000:: *53C21 Methods of Riemannian geometry (global)
53C44 Geometric evolution equations (mean curvature flow)
35K55 Nonlinear parabolic equations

Keywords: Ricci-curvature; equation of evolution; Nash-Moser inverse function theorem; method of a priori estimation; constant positive curvature

Citations: Zbl 0162.53304; Zbl 0437.53011

Cited in: Zbl 1215.53062 Zbl 1216.53057 Zbl 1170.53044 Zbl 1160.53368 Zbl 1204.53033 Zbl 1163.53042 Zbl 1157.53002 Zbl 1157.53035 Zbl 1148.53050 Zbl 1162.53026 Zbl 1157.53034 Zbl 1200.53057 Zbl 1105.58013 Zbl 1091.53043 Zbl 1086.53085 Zbl 1067.53053 Zbl 1130.53002 Zbl 1108.53002 Zbl 1130.53001 Zbl 1044.53049 Zbl 0985.53037 Zbl 0999.53041 Zbl 0924.53038 Zbl 0896.53003 Zbl 0894.53042 Zbl 0879.53024 Zbl 0867.53031 Zbl 0799.53050 Zbl 0890.53007 Zbl 0752.53025 Zbl 0725.53050 Zbl 0668.53026 Zbl 0675.53044 Zbl 0646.53073 Zbl 0642.53071 Zbl 0646.06010 Zbl 0642.53074 Zbl 0631.53028 Zbl 0613.53018 Zbl 0607.53027 Zbl 0589.53046 Zbl 0599.53037 Zbl 0556.53001 Zbl 0547.53034 Zbl 0517.53044

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