Guofang Wei

Analysis and geometry on manifolds with integral Ricci curvature bounds. II

We extend several geometrical results for manifolds with lower Ricci curvature bounds to situations where one has integral lower bounds. In particular we generalize Colding’s volume convergence results and extend the Cheeger-Colding splitting theorem.

Hausdorff Convergence and Universal Covers

We prove that if Y is the Gromov-Hausdorff limit of a sequence of compact manifolds, M n i , with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then Y has a universal cover. We then show that, for i sufficiently large, the fundamental group of Mi has a surjective homeomorphism onto the group of deck transforms of Y . Finally, in the non-collapsed case where the Mi have an additional uniform lower bound on volume, we prove that the kernels of these surjective maps are finite with a uniform bound on their cardinality. A number of theorems are also proven concerning the limits of covering spaces and their deck transforms when the Mi are only assumed to be compact length spaces with a uniform upper bound on diameter.

Smoothing Riemannian Metrics with Ricci Curvature Bounds

We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds.

Universal covers for Hausdorff limits of noncompact spaces

We prove that if Y is the Gromov-Hausdorff limit of a sequence of complete manifolds, M n i , with a uniform lower bound on Ricci curvature, then Y has a universal cover.

On the fundamental groups of manifolds with almost-nonnegative Ricci curvature

We give an upper bound on the growth of 7rI (M) for a class of manifolds M with Ricci curvature RicM > -c, diameter d(M) 1 and volume vol(M) > v. In [4], Milnor proved that every finitely generated subgroup of the fundamental group of a manifold Mn with nonnegative Ricci curvature is of polynomial growth with degree 0, there exists E = e(n, v) > 0 such that if a complete manifold Mn admits a metric satisfying the conditions RicM, -6, d(M) =1, and vol(M) > v, then thefundamental group of M is of polynomial growth with degree (n 1)H, vol(M) > v, and d(M) < D, there are only finitely many isomorphism classes of r I(M). Proof of Theorem 1. Choose a base point x in the universial covering M A M, and let x0 = p(.0) and g1, . ? . , gr be a set of generators of the fundamental group 7c1 (M) viewed as deck transformations in the isometry group of M. Denote F(s) { distinct words in 7, (M) of length < s}, y(s) = #F(s), and I = maxl<i<r{d(o then U g (F) c Bsl+d () geF(S) where d =d (M) 1 . Therefore, (1) y (s) * vol (M) ? vol (Bsl+ 1 (JO)) Received by the editors March 13, 1989 and, in revised form, October 23, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C20; Secondary 57S20. This research was supported in part by Alfred P. Sloan Doctoral Dissertation Fellowship. ? 1990 American Mathematical Society 0002-9939/90

Metrics of positive Ricci curvature on bundles

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Ricci Curvature and Betti Numbers

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Negative Ricci curvature and isometry group

We construct new examples of manifolds of positive Ricci curvature which, topologically, are vector bundles over compact manifolds of almost nonnegative Ricci curvature. In particular, we prove that if E is the total space of a vector bundle over a compact manifold of nonnegative Ricci curvature, then E × ℝ p admits a complete metric of positive Ricci curvature for all large p.

On volumes of hyperbolic orbifolds

We derive a uniform bound for the total betti number of a closed manifold in terms of a Ricci curvature lower bound, a conjugate radius lower bound and a diameter upper bound. The result is based on an angle version of Toponogov comparison estimate for small triangles in a complete manifold with a Ricci curvature lower bound. We also give a uniform estimate on the generators of the fundamental group and prove a fibration theorem in this setting.

The cut-off covering spectrum

We show that for n-dimensional manifolds with Ricci curvature bounded between two negative constants the order of their isometry groups is uniformly bounded by the Ricci curvature bounds, the volume, and the injectivity radius. We also show that the degree of symmetry (seex2 for denition) is lower semicontinuous in the Gromov-Hausdor topology.