Related papers: Matrix Inequality for the Laplace Equation

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…

We prove an estimate for solutions to the linearized Ricci flow system on closed 3-manifolds. This estimate is a generalization of Hamilton's pinching is preserved estimate for the Ricci curvatures of solutions to the Ricci flow on…

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…

We first prove a uniform integral Laplace comparison result for the K\"ahler Ricci flow on Fano manifolds which depends only on the initial metric. As an application, using Cheeger-Colding theory and previous results by some of the authors,…

Let $(M,g(t))$, $0\le t\le T$, $\partial M\ne\phi$, be a compact $n$-dimensional manifold, $n\ge 2$, with metric $g(t)$ evolving by the Ricci flow such that the second fundamental form of $\partial M$ with respect to the unit outward normal…

We prove the Li-Yau gradient estimate for the heat kernel on graphs. The only assumption is a variant of the curvature-dimension inequality, which is purely local, and can be considered as a new notion of curvature for graphs. We compute…

We establish an estimate for the fundamental solution of the heat equation on a closed Riemannian manifold $M$ of dimension at least 3, evolving under the Ricci flow. The estimate depends on some constants arising from a Sobolev imbedding…

Inspired by Hongjie Dong and Qi S. Zhang's article \cite{ZQ2}, we find that the analyticity in time for a smooth solution of the heat equation with exponential quadratic growth in the space variable can be extended to any complete…

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 article we derive Harnack estimates for conjugate heat kernel in an abstract geometric flow. Our calculation involves a correction term D. When D is nonnegative, we are able to obtain a Harnack inequality. Our abstract formulation…

We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the $L^2$-space produces the same evolution as the gradient flow of the relative entropy in the $L^2$-Wasserstein space.…

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 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…

We prove a sharp gradient estimate for the natural Green's function of a closed manifold with positive Ricci curvature. We also show that this estimate is closely related to a family of monotonicity formulae. These results extend those…

We estimate the heat kernel on a closed Riemannian manifold $M$, with $dim(M)\geq 3$, evolving under the Ricci-harmonic map flow and the result depends on some constants arising from a Sobolev imbedding theorem. In a special case, when the…

Uniform $L^1$ and lower bounds are obtained for the Green's function on compact K\"ahler manifolds. Unlike in the classic theorem of Cheng-Li for Riemannian manifolds, the lower bounds do not depend directly on the Ricci curvature, but only…

An explicit representation formula for all positive ancient solutions of the heat equation in the Euclidean case is found. In the Riemannian case with nonnegative Ricci curvature, a similar but less explicit formula is also found. Here it…

Two sharp comparison results are derived for three-dimensional complete noncompact manifolds with scalar curvature bounded from below. The first one concerns the Green's function. When the scalar curvature is nonnegative, it states that the…

We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation $\omega_t=\Delta\omega+a\omega\ln \omega$ on closed manifolds. We also derive a new interpolated Harnack inequality for the…

We study the Ricci-Bourguignon flow on warped product manifolds with noncompact base. This setting leads naturally to a parabolic partial differential equation on the space of smooth warping functions, arising from the necessary and…