On Complete Manifolds With Nonnegative Ricci Curvature

Abstract

ON COMPLETE MANIFOLDS WITH NONNEGATIVE RICCI CURVATURE UWE ABRESCH AND DETLEF GROMOLL Complete open Riemannian manifolds (M n , g) with nonnegative sectional curvature are well understood. The basic results are Toponogov's Splitting Theorem and the Soul Theorem [CGI]. The Splitting Theorem has been extended to manifolds of nonnegative Ricci curvature [CG2]. On the other hand, the Soul Theorem does not extend even topologically, according to recent examples in [GM2]. A different method to construct manifolds which carry a metric with Ric> 0, but no metric with nonnegative sectional curvature, has been given by L. Berard Bergery [BB]. This leads to the question (cf. also [YI]): Is there any finiteness result for complete Riemannian manifolds with Ric ~ O? The answer is certainly affirmative in the low-dimensional special cases n = 2, where all notions of curvature coincide, and n = 3, where nonnegative Ricci curvature has been studied by means of stable minimal surfaces [MSY, SY]. On the other hand, J. P. Sha and D. G. Yang [ShY] have constructed complete manifolds with strictly positive Ricci curvature in higher dimensions. For example they can choose the underlying space to be R4 x S3 with infinitely many copies of S3 x Cp2 attached to it by surgery; cf. also [ShYI]. It is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions. The purpose of this paper is to establish the following main result. Theorem A. Let M n be a complete open Riemannian manifold with Ric ~ O. Suppose that M n has diameter growth of order o(rl/n). Then M n is homotopy equivalent to the interior of a compact manifold with boundary, provided the sectional curvature is bounded away from-00. The notion of diameter growth requires a precise definition. Roughly speaking, we would like to measure the diameters of the "essential components" of the distance spheres S(po' r) W.r.t. the intrinsic metric in Mn\B(po' ,. r) , where ! < , < I is a fixed number. Given any open set Q c M n , not necessarily connected, we shall write diam(L, Q) for the diameter of any connected subset L c Q measured W.r.t. the intrinsic distance function of the open submanifold

Book Review: Metric structures for Riemannian and non-Riemannian spaces

“…The main breakthrough came with Colding's L 2 − average version of Toponogov's triangle comparison theorem for thin triangles [17]. Prior to that Abresch and Gromoll [1] had obtained an estimate for the excess (failure of triangle inequality from equality) of thin triangles. This delicate estimate can be viewed as a weakened finite quantitive version of the Cheeger-Gromoll splitting theorem [15], asserting that a line (geodesic which is minimal between any two of its points) splits off isometrically in a complete manifold of nonnegative Ricci curvature.…”

Ball covering on manifolds with nonnegative Ricci curvature near infinity

“…We omit the details. Since (a) implies (b) (see [AG,Proposition 4.3 and its proof]) which in turn implies (c), let us assume that the end E satisfies condition (c). The proposition is a consequence of the following two lemmas.…”

Solutions of a Certain Nonlinear Elliptic Equation on Riemannian Manifolds

“…A function u in W 1,p loc (Ω) is a solution (supersolution, subsolution, respectively) of the equation − divA x (∇u) = 0 (≥ 0, ≤ 0, respectively) (1) in Ω if Ω A x (∇u), ∇φ = 0 (≥ 0, ≤ 0, respectively) for any (nonnegative, respectively) φ ∈ C ∞ 0 (Ω). We say that a function u is A-harmonic (of type p) if u is a continuous solution of the equation (1).…”

“…We say that a function u is A-harmonic (of type p) if u is a continuous solution of the equation (1). In the typical case A x (ξ) = ξ|ξ| p−2 , A-harmonic functions are called p-harmonic and, in particular, if p = 2, we obtain harmonic functions.…”

“…Furthermore, for each end E of M , ∂B R (o) ∩ E is connected, where o is a fixed point of M . (See [1].) Since M satisfies the volume doubling condition globally, the finite covering condition (FC) is valid on each end.…”