Abstract. In this paper, we give a complete proof of the Poincaré and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.

No abstract

In this paper we prove that there is no κ-solution of Ricci flow on 3-dimensional noncompact manifold with strictly positive sectional curvature and blow up at some finite time T satisfying T 0 √ T − tR(p 0 , t)dt < ∞ for some point p 0 . This partially confirms a conjecture of Perelman.2000 Mathematics Subject Classification. Primary 53C44; Secondary 53C42, 57M50.

No abstract

In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.1 A metric measure space is a metric space equipped a Borel measure. 1 2 HUI-CHUN ZHANG AND XI-PING ZHU a slight modification of a property introduced earlier by Sturm in [S3] and in a similar form by Kuwae and Shioya in [KS3,KS4]. The condition M CP (n, k) is indeed an infinitesimal version of the Bishop-Gromov relative volume comparison condition. For a metric measure space, Sturm [S2] proved that CD(n, k) implies M CP (n, k) provided it is non-branching 2 . Note that any Alexandrov space with curvature bounded below is non-branching. Recently, Petrunin [Pet2] proved that any n-dimensional Alexandrov space with curvature 0 must satisfy CD(n, 0) and claimed the general statement that the condition curvature k (for some k ∈ R) implies the condition CD(n, (n − 1)k) can be also proved along the same lines.Let M be a Riemannian manifold with Riemannian distance d and Riemannian volume vol. Lott, Villani in [LV1] and von Renesse, Sturm in [RS, S4] proved that (M, d, vol) satisfies CD(∞, k) if and only if Ric(M ) k. Indeed, they proved a stronger weighted version (see Theorem 7.3 in [LV1] and Theorem 1.1 in [RS], Theorem 1.3 in [S4]). Let φ be a smooth function on M with M e −φ dvol = 1. Lott and Villani in [LV2] proved that (M, d, e −φ · vol) satisfies CD(n, k) if and only if weighted Ricci curvature Ric n (M ) k (see Definition 4.20-the definition of Ric n -and Theorem 4.22 in [LV2]). A similar result was proved by Sturm in [S2] (see Theorem 1.7 in [S2]). In particular, they proved that (M, d, vol) satisfies CD(n, k) if and only if Ric(M ) k and dim(M ) n. If dim(M ) = n, Ohta in [O1] and Sturm in [S2] proved, independently, that M satisfies M CP (n, k) is equivalent to Ric(M ) k.Nevertheless, since n-dimensional norm spaces (V n , · p ) satisfy CD(n, 0) for every p > 1 (see, for example, page 892 in [V]), it is impossible to show Cheeger-Gromoll splitting theorem under CD(n, 0) for general metric measure spaces. Furthermore, it was shown by Ohta in [O3] that on a Finsler manifolds M , the curvature-dimension condition CD(n, k) is equivalent to the weighted Finsler Ricci curvature condition Ric n (M ) k (see also [O4] or [OSt], refer to [O4] for the definition Ric n in Finsler manifolds). That says, the curvaturedimension condition is somewhat a Finsler geometry character. Seemly, it is difficult to show the rigidity theorems, such as Cheng's maximal diameter theorem and Obata's theorem, under CD(n, n − 1) for general metric measure spaces.As a compensation, Watanabe [W] proved that if a metric measure space M satisfies CD(n, 0) or M CP (n, 0) then M has at most two ends. Ohta [O2] proved that a nonbranching compact metric measure space with M CP (n, n − 1) and diameter = π is homeomorphic to a spherical suspension.Alexandrov spaces with curvature bounded below have richer geomet...