Related papers: A direct approach to sharp Li-Yau Estimates on clo…

In this paper, motivated by the works of Bakry et. al in finding sharp Li-Yau type gradient estimate for positive solutions of the heat equation on complete Riemannian manifolds with nonzero Ricci curvature lower bound, we first introduce a…

In this paper, motivated by finding sharp Li-Yau type gradient estimate for positive solution of heat equations on complete Riemannian manifolds with negative Ricci curvature lower bound, we first introduce the notion of Li-Yau multiplier…

The paper considers a manifold $M$ evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on $M$. Among other results, we prove Li-Yau-type inequalities in this context. We…

In this paper, we remove the assumption on the gradient of the Ricci curvature in Hamilton's matrix Harnack estimate for the heat equation on all closed manifolds, answering a question which has been around since the 1990s. New ingredients…

We prove Li-Yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the Ricci flow. In the former case the curvature condition is $|Ric^-| \in L^p$ for some $p>n/2$, or $\sup_\M \int_\M…

In 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…

The paper establishes a series of gradient estimates for positive solutions to the heat equation on a manifold $M$ evolving under the Ricci flow, coupled with the harmonic map flow between $M$ and a second manifold $N$. We prove Li-Yau type…

In this paper we extend a gradient estimate of R. Hamilton for positive solutions to the heat equation on closed manifolds to bounded positive solutions on complete, non-compact manifolds with $Rc \geq -Kg$. We accomplish this extension via…

We prove a Li-Yau gradient estimate for positive solutions to the heat equation, with Neumann boundary conditions, on a compact Riemannian submanifold with boundary ${\bf M}^n\subseteq {\bf N}^n$, satisfying the integral Ricci curvature…

Let $(\M^n, g)$ be a $n$ dimensional, complete ( compact or noncompact) Riemannian manifold whose Ricci curvature is bounded from below by a constant $-K \le 0$. Let $u$ be a positive solution of the heat equation on $\M^n \times (0,…

We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for…

In this paper, motivated by the work of Qi S. Zhang in [28], we derive Li-Yau gradient bounds for positive solutions of the f-heat equation on closed manifolds with Bakry-Emery Ricci curvature bounded below.

We derive localized and global noncompact versions of Hamilton's gradient estimate for positive solutions to the heat equation on Riemannian manifolds with Ricci curvature bounded below. Our estimates are essentially optimal and…

In this paper we prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply such estimates to establish the monotonicity of…

In this paper, by maximum principle and cutoff function, we investigate gradient estimates for positive solutions to two nonlinear parabolic equations under Ricci flow. The related Harnack inequalities are deduced. An result about positive…

In this paper we give Hamilton's Laplacian estimates for the heat equation on complete noncompact manifolds with nonnegative Ricci curvature. As an application, combining Li-Yau's lower and upper bounds of the heat kernel, we give an…

In this short survey paper, we first recall the log gradient estimates for the heat equation on manifolds by Li-Yau, R. Hamilton and later by Perelman in conjunction with the Ricci flow. Then we will discuss some of their applications and…

In this paper, we will establish an elliptic local Li-Yau gradient estimate for weak solutions of the heat equation on metric measure spaces with generalized Ricci curvature bounded from below. One of its main applications is a sharp…

In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem $$ u_t-\Delta u=au\log u, \ \ u>0 $$ on the compact Riemannian…

We derive a gradient estimate for positive functions, in particular for positive solutions to the heat equation, on finite or locally finite graphs. Unlike the well known Li-Yau estimate, which is based on the maximum principle, our…