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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.

In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau’s estimate for weak solutions of the heat equation and prove a sharp Yau’s gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition RCD(K; N).

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 this paper, we extend the sharp lower bounds of spectal gap, due to Chen- Wang [12, 13], Bakry-Qian [6] and Andrews-Clutterbuck [5], from smooth Riemannian manifolds to general metric measure spaces with Riemannian curvature-dimension condition RCD(K;N).

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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.

In this paper, we will prove the Weyl's law for the asymptotic formula of Dirichlet eigenvalues on metric measure spaces with generalized Ricci curvature bounded from below.

In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space (X, dX ) with curvature bounded above by a constant κ (κ 􏰣 0) in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng (1980) and Choi (1982) to harmonic maps into singular spaces.

Let M be a complete Riemannian manifold with Riemannian volume volg and f be a smooth function on M. A sharp upper bound estimate on the first eigenvalue of symmetric diffusion operator 􏰂 f = 􏰂 − ∇ f · ∇ was given by Wu (J Math Anal Appl 361:10–18, 2010) and Wang (Ann Glob Anal Geom 37:393–402, 2010) under a condition that finite dimensional Bakry–Émery Ricci curvature is bounded below, independently. They propounded an open problem is whether there is some rigidity on the estimate. In this note, we will solve this problem to obtain a splitting type theorem, which generalizes Li–Wang’s result in (J Differ Geom 58:501–534, 2001, J Differ Geom 62:143–162, 2002). For the case that infinite dimensional Bakry–Emery Ricci curvature of M is bounded below, we donotexpectanyupperboundestimateonthefirsteigenvalueof􏰂f withoutanyadditional assumption (see the example in Sect. 2). In this case, we will give a sharp upper bound estimate on the first eigenvalue of􏰂f under the additional assuption that |∇f| is bounded.We also obtain the rigidity result on this estimate, as another Li–Wang type splitting theorem.

In this paper, we shall discuss the existence, uniqueness and regularity of harmonic maps from an Alexandrov space into a geodesic space with curvature 􏰢 1 in the sense of Alexandrov.