In this paper, we derive some local a priori estimates for the Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow on R3, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t) ≡ E.

We prove exponential estimates for plurisubharmonic functions with respect to Monge-Amp`ere measures with H¨older continuous potential. As an application, we obtain several stochastic properties for the equilibrium measures associated to holomorphic maps on projective spaces. More precisely, we prove the exponential decay of correlations, the central limit theorem for general d.s.h. observables, and the large deviations theorem for bounded d.s.h. observables and H¨older continuous observables.

In this paper, we completely classify all compact 4-manifolds with positive isotropic curvature. We show that they are diffeomorphic to S4 or RP4 or quotients of S3 ×R by a cocompact fixed point free subgroup of the isometry group of the standard metric of S3 × R, or a connected sum of them.

In this paper, we prove some analogues of Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet (discrete) Laplace eigenvalues on any subset in the integer lattice $\Z^n.$ This partially answers a question posed by Chung and Oden.

In this paper, we first derive the CR analogue of matrix Li–Yau–Hamilton inequality for the positive solution to the CR heat equation in a closed pseudohermitian (2n + 1)-manifold with nonnegative bisectional curvature and bitorsional tensor. We then obtain the CR Li–Yau gradient estimate in the Heisenberg group. We apply this CR gradient estimate and extend the CR matrix Li–Yau–Hamilton inequality to the case of the Heisenberg group. As a consequence, we derive the Hessian comparison property for the Heisenberg group.