An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichm¨uller and moduli spaces. The tensor is evaluated on the gradients of geodesiclengths for disjoint geodesics. A precise lower bound for sectional curvature in terms of the surface systole is presented. The curvature tensor expansion is applied to establish continuity properties at the frontier strata of the augmented Teichm¨uller space. The curvature tensor has the asymptotic product structure already observed for the metric and covariant derivative. The product structure is combined with the earlier negative sectional curvature results to establish a classification of asymptotic flats. Furthermore, tangent subspaces of more than half the dimension of Teichm¨uller space contain sections with a definite amount of negative curvature. Proofs combine estimates for uniformization group exponential-distance sums and potential theory bounds.

An odd Seiberg-Witten invariant imposes bounds on the signature of a closed, almost complex 4-manifold with vanishing first Chern class. This applies in particular to symplectic 4-manifolds of Kodaira dimension zero.

In this paper we survey certain aspects of the classical Weil-Petersson metric and its generalizations. Being a natural L^2 metric on the parameter space of a family of complex manifolds or holomorphic vector bundles which admit some canonical metrics, the Weil-Petersson metric is well defined when the automorphism group of each fiber is discrete and the curvature of the Weil-Petersson metric can be computed via certain integrals over each fiber. We will discuss the case when these fibers have continuous automorphism groups. We also discuss the relation between the Weil-Petersson metric and energy of harmonic maps.

We define the notion of a Ricci curvature lower bound for parametrized statistical models. Following the seminal ideas of Lott–Sturm–Villani, we define this notion based on the geodesic convexity of the Kullback–Leibler divergence in a Wasserstein statistical manifold, that is, a manifold of probability distributions endowed with a Wasserstein metric tensor structure. Within these definitions, which are based on Fisher information matrix and Wasserstein Christoffel symbols, the Ricci curvature is related to both, information geometry and Wasserstein geometry. These definitions allow us to formulate bounds on the convergence rate of Wasserstein gradient flows and information functional inequalities in parameter space. We discuss examples of Ricci curvature lower bounds and convergence rates in exponential family models.

In this paper, we derive two subgradient estimates of the CR heat equation in a closed pseudohermitian 3-manifold which are served as the CR version of the Li-Yau gradient estimate. With its applications, we first get a subgradient estimate of the logarithm of the positive solution of the CR heat equation. Secondly, we have the Harnack inequality and upper bound estimate for the heat kernel. Finally, we obtain Perelman-type entropy formulae for the CR heat equation.