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...

In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yau's gradient estimate for harmonic functions is also obtained on Alexandrov spaces.

In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. In [39], F. H. Lin proposed an open problem: Can the Hölder continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in [28]). The main theorem of this paper gives a complete resolution to it.

Let (X, d, µ) be a RCD * (K, N) space with K ∈ R and N ∈ [1, ∞). We derive the upper and lower bounds of the heat kernel on (X, d, µ) by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the L p boundedness of (local) Riesz transforms.

In the previous work [35], the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen-Wang [9, 10] and Bakry-Qian [6], from smooth Riemannian manifolds to Alexandrov spaces. As an application, we get an Obata type theorem for Alexandrov spaces. Secondly, we obtain (sharp) Li-Yau's estimate for positve solutions of heat equations on Alexandrov spaces.