One of the main purposes of this paper is to understand the geometry of the moduli and the Teichmller spaces of Riemann surfaces. The most interesting results we have in this paper are the detailed understanding of two new complete Khler metrics with nice properties and the KhlerEinstein metric on the Teichmller and the moduli spaces of Riemann surfaces. The two new metrics, the Ricci metric and the perturbed Ricci metric, are naturally defined as the negative Ricci curvature of the WeilPetersson metric and a combination of it with the WeilPetersson metric. We prove that these new metrics and the KhlerEinstein metric on the Teichmller and moduli spaces all have Poincar type boundary behavior, and further, in [7], we prove that they all have bounded geometry. Note that the KhlerEinstein metric is the key link between the differential geometric and algebraic geometric aspects of these spaces

We derive a gradient estimate for positive functions, in particular for positive solutions to the heat equation, on finite or locally finite graphs. Unlike the well known Li-Yau estimate, which is based on the maximum principle, our estimate follows from the graph structure of the gradient form and the Laplacian operator. Though our assumption on graphs is slightly stronger than that of Bauer et al. (J Differ Geom 99:359–405, 2015), our estimate can be easily applied to nonlinear differential equations, as well as differential inequalities. As applications, we estimate the greatest lower bound of Cheng’s eigenvalue and an upper bound of the minimal heat kernel, which is recently studied by Bauer et al. (Preprint, 2015) by the Li-Yau estimate. Moreover, generalizing an earlier result of Lin and Yau (Math Res Lett 17:343–356, 2010), we derive a lower bound of nonzero eigenvalues by our gradient estimate.

We determine regularity results for energy minimizing maps from an n-dimensional Riemannian polyhedral complex X into a CAT(1) space. Provided that the metric on X is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the (n−2)-skeleton, we improve the regularity to locally Lipschitz. Finally, for points x ∈ X^{(k)} with k ≤ n−2, we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of x ∈ X.

Continuing our previous work (arXiv:1509.07981v1), we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs. These global gradient estimates can be compared with the Li–Yau inequality on graphs contributed by Bauer et al. [J. Differential Geom., 99, 359–409 (2015)]. In many topics, such as eigenvalue estimate and heat kernel estimate (not including the Liouville type theorems), replacing the Li–Yau inequality by the global gradient estimate, we can get similar results.

In this paper, we prove that, a compact complex manifold $X$ admits a smooth Hermitian metric with positive (resp., negative) scalar curvature if and only if $K_X$ (resp., $K_X^{-1}$) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold $X$ with complex dimension $\geq 2$, there exist smooth Hermitian metrics with positive total scalar curvature, and one of the key ingredients in the proof relies on a recent solution to the Gauduchon conjecture by G. Székelyhidi, V. Tosatti, and B. Weinkove.