Sharp Li–Yau-Type Gradient Estimates on Hyperbolic Spaces

In this paper, motivated by the works of Bakry et al. in finding sharp Li–Yau-type gradient estimates for positive solutions of the heat equation on complete Riemannian manifolds with nonzero Ricci curvature lower bound, we first introduce a general form of Li–Yau-type gradient estimate and show that the validity of such an estimate for any positive solutions of the heat equation reduces to the validity of the estimate for the heat kernel of the Riemannian manifold. Then, a sharp Li–Yau-type gradient estimate on the three-dimensional hyperbolic space is obtained by using the explicit expression of the heat kernel, and some optimal Li–Yau-type gradient estimates on general hyperbolic spaces are obtained.