A NOTE ON LI-YAU-TYPE GRADIENT ESTIMATE

In this article, we obtain Li-Yau-type gradient estimates with time dependent parameter for positive solutions of the heat equation that are different with the estimates by Li-Xu [21] and Qian [23]. As an application of the estimate, we also obtained slight improvements of Davies' Li-Yau-type gradient estimate.     

DE RHAM DECOMPOSITION FOR RIEMANNIAN MANIFOLDS WITH BOUNDARY

In this paper,we extend the classical de Rham decomposition theorem to the case of Riemannian manifolds with boundary by using the trick of the development of curves.