Ricci Curvature on Alexandrov spaces and Rigidity Theorems

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.Comment: final versio

Lipschitz continuity of harmonic maps between 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 H\"older continuous. In [39], F. H. Lin proposed a challenge problem: Can the H\"older 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.Comment: We remove the assumption in the previous version that the domain space has nonnegative generalized Ricci curvature. This solves Lin's conjecture completely. To appear in Invent. Mat

Sharp Spectral Gap and Li-Yau's Estimate on Alexandrov Spaces

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.Comment: 19 pages, final version, to appear in Math.