Li–Yau–Hamilton estimates and Bakry–Emery–Ricci curvature

Abstract

a b s t r a c tIn this paper we derive Cheng-Yau, Li-Yau, Hamilton estimates for Riemannian manifolds with Bakry-Emery-Ricci curvature bounded from below, and also global and local upper bounds, in terms of Bakry-Emery-Ricci curvature, for the Hessian of positive and bounded solutions of the weighted heat equation on a closed Riemannian manifold.

Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds

“…Recently, Dung and the author investigated gradient estimates of Hamilton-Souplet-Zhang type. Our work is a generalization of the results of Huang-Ma, Y. Li and other mathematicians, see [3,5,6] for further discussion and the references there in. Motivated by the above result, it is very natural for us to look for gradient estimates of Li-Yau type for the general heat equation (1.1).…”

“…where ∇ and ∆ are respectively the Levi-Civita connection and the Laplace-Beltrami operator with respect to g, V is a smooth vector field on M . In [1] and [6], the authors introduced two curvatures…”

Gradient Estimates for a Nonlinear Heat Equation Under Finsler-Geometric Flow

“…Therefore, Li-Yau inequality is often called differential Harnack inequality. Li-Yau type gradient estimates have been obtained for other nonlinear equations on manifolds, see for example [4][5][6][7][8][9][10][11][12][13] and the references therein. Over the past two decades, many authors used similar techniques to prove gradient estimates and Harnack inequalities for geometric flows.…”

Gradient estimates of Hamilton–Souplet–Zhang type for a general heat equation on Riemannian manifolds

“…V is a smooth vector field on M . A natural generalization of Bakry-Émery curvature and N -Bakry-Émery curvature are the following two tensors ( [3,7])…”

“…When V = ∇f and f is a smooth function on M then Ric V , Ric N V become Bakry-Émery curvature and N -BakryEmery curvature. In [7], Li studied gradient estimates of Li-Yau and Hamilton type for the following general heat equation…”