Related papers: Matrix Li-Yau-Hamilton Estimates for Nonlinear Hea…
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 prove a new matrix Li-Yau-Hamilton estimate for K\"ahler-Ricci flow. The form of this new Li-Yau-Hamilton estimate is obtained by the interpolation consideration originated in \cite{Ch1}. This new inequality is shown to be…
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…
We generalize Hamilton's matrix Li-Yau-type Harnack estimate for the Ricci flow by considering the space of all LYH (Li-Yau-Hamilton) quadratics that arise as curvature tensors of space-time connections satisfying the Ricci flow with…
In this paper, we derive local and global Li-Yau type gradient estimates for the positive solutions of the CR heat equation on complete noncompact pseudo-Hermitian manifolds. As applications of the gradient estimates, we give a Harnack…
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…
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…
Recently, Qi S.Zhang [26] has derived a sharp Li-Yau estimate for positive solutions of the heat equation on closed Riemannian manifolds with the Ricci curvature bounded below by a negative constant. The proof is based on an integral…
In this paper, we establish a priori estimates and existence results for solutions of a general class of fully non-linear equations on noncompact K\"{a}hler and Hermitian manifolds. As geometric applications, we construct complete…
This work deals with the Entire solutions of a nonlinear equation. The first part of this paper is devoted to investigation of the Liouville property on compact manifolds, which extends a result by Castorina-Mantegazza [4] for positive f.…
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…
We calculate heat invariants of arbitrary Riemannian manifolds without boundary. Every heat invariant is expressed in terms of powers of the Laplacian and the distance function. Our approach is based on a multi-dimensional generalization of…
The main goal of this paper is to generalize some Li-Yau type gradient estimates to Finsler geometry in order to derive Harnack type inequalities. Moreover, we obtain, under some curvature assumption, a general gradient estimate for…
In this paper, we study elliptic gradient estimates for a nonlinear $f$-heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton's and…
Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by a geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors. We derive several…
We study the nonlinear generalized heat equation $C(u)u_t=\frac{1}{z^{\nu}}\left(K(u)z^{\nu}u_z\right)_z$, where $C(u)$ and $K(u)$ are temperature-dependent thermal coefficients and $\nu>0$ is a geometric parameter describing the varying…
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…
In this note, we prove some new entropy formula for linear heat equation on static Riemannian manifold with nonnegative Ricci curvature. The results are analogies of Cao and Hamilton's entropies for Ricci flow coupled with heat-type…
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…
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…