We prove some structure results for transversely reducible Sasaki manifolds. In particular, we show a Sasaki manifold with positive Ricci curvature is transversely irreducible, and so join (product) construction cannot be generalized to irregular SasakiEinstein manifolds, as opposed to the quasi-regular case done by WangZiller and BoyerGalicki. As an application, we classify compact Sasaki manifolds with nonnegative transverse bisectional curvature, which can be viewed as the generalized Frankel conjecture in Sasaki geometry.

In this paper, we show that any compact Kähler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a Kähler–Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold X homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure J compatible with the symplectic form, there is no non-constant J-holomorphic entire curve f:C→X.

We prove several vanishing theorems for a class of generalized elliptic genera on foliated manifolds, by using classical equivariant index theory. The main techniques are the use of the Jacobi theta-functions and the construction of a new class of elliptic operators associated to foliations.

In this paper, we generalize the Cao-Yau'’s gradient estimate for the sum of squares of vector …elds up to higher step under ssumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As consequences, we obtain the Harnack inequality and upper bound estimate for the CR heat kernel. 1.

In this paper, we continue our study of the canonical metrics on the moduli space of curves. We first prove the bounded geometry of the Ricci and perturbed Ricci metrics. By carefully choosing the pertubation constant and by studying the asymptotics, we show that the Ricci and holomorphic sectional curvatures of the perturbed Ricci metric are bounded from above and below by negative constants. Based on our understanding of the KhlerEinstein metric, we show that the logarithmic cotangent bundle over the DeligneMumford moduli space is stable with respect to the canonical polarization. Finally, in the last section, we prove the strongly bounded geometry of the KhlerEinstein metric by using the KhlerRicci flow and a priori estimates of the complex Monge-Ampere equation.