scholarly journals A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow

2008 ◽  
Vol 255 (4) ◽  
pp. 1008-1023 ◽  
Author(s):  
Shilong Kuang ◽  
Qi S. Zhang
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Ricci Flow ◽  
Gradient Estimate ◽  
Conjugate Heat Equation

Gradient estimate for the forward conjugate heat equation of forms under the Ricci flow

2021 ◽  
pp. 2150081
Author(s):  
Liangdi Zhang
Keyword(s):  
Riemannian Manifold ◽  
Heat Equation ◽  
Ricci Flow ◽  
Differential Forms ◽  
Gradient Estimate ◽  
Conjugate Heat Equation

We establish bounds for the gradient of solutions to the forward conjugate heat equation of differential forms on a Riemannian manifold with the metric evolves under the Ricci flow.

Gradient and Harnack-type estimates for PageRank

2020 ◽  
pp. 1-19
Author(s):  
Paul Horn ◽  
Lauren M. Nelsen
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Gradient Estimate ◽  
Personalized Pagerank ◽  
Geometric Properties ◽  
Algorithmic Design

Abstract Personalized PageRank has found many uses in not only the ranking of webpages, but also algorithmic design, due to its ability to capture certain geometric properties of networks. In this paper, we study the diffusion of PageRank: how varying the jumping (or teleportation) constant affects PageRank values. To this end, we prove a gradient estimate for PageRank, akin to the Li–Yau inequality for positive solutions to the heat equation (for manifolds, with later versions adapted to graphs).

scholarly journals Gradient estimates for the heat equation under the Ricci-harmonic map flow

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Mihai Bailesteanu
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Ricci Flow ◽  
Harmonic Map ◽  
Gradient Estimates ◽  
Complete Manifold ◽  
Harnack Inequalities ◽  
Harmonic Map Flow

AbstractThe 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 Harnack inequalities and we consider the cases when M is a complete manifold without boundary and when M is compact without boundary.

Gradient estimates for a nonlinear heat equation under the Finsler-Ricci flow

2019 ◽  
Vol 69 (2) ◽  
pp. 409-424
Author(s):  
Fanqi Zeng ◽  
Qun He
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Ricci Flow ◽  
Finsler Manifold ◽  
Nonlinear Heat Equation ◽  
Gradient Estimates ◽  
Harnack Inequalities

Abstract This paper considers a compact Finsler manifold (Mn, F(t), m) evolving under the Finsler-Ricci flow and establishes global gradient estimates for positive solutions of the following nonlinear heat equation: $$\begin{array}{} \partial_{t}u=\Delta_{m} u, \end{array} $$ where Δm is the Finsler-Laplacian. As applications, several Harnack inequalities are obtained.

scholarly journals Differential Harnack estimates for conjugate heat equation under the Ricci flow

2015 ◽  
Vol 08 (04) ◽  
pp. 1550063
Author(s):  
Abimbola Abolarinwa
Keyword(s):  
Heat Equation ◽  
Ricci Flow ◽  
Beltrami Operator ◽  
Nonlinear Heat Equation ◽  
Curvature Operator ◽  
Harnack Estimate ◽  
Diffusion Operator ◽  
Harnack Inequalities ◽  
Harnack Estimates ◽  
Conjugate Heat Equation

We prove (local and global) differential Harnack inequalities for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace–Beltrami operator is replaced with “[Formula: see text]”, where [Formula: see text] is the scalar curvature of the Ricci flow, which is well generalized to the case of nonlinear heat equation with potential. Our estimates improve on some well known results by weakening the curvature constraints. As a by-product, we obtain some Li–Yau-type differential Harnack estimate. The localized version of our estimate is very useful in extending the results obtained to noncompact case.

scholarly journals Matrix Inequality for the Laplace Equation

2017 ◽  
Vol 2019 (11) ◽  
pp. 3485-3497
Author(s):  
Jiewon Park
Keyword(s):  
Heat Equation ◽  
Ricci Flow ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Volume Growth ◽  
Curvature Flow ◽  
Gradient Estimate ◽  
Matrix Inequality ◽  
Nonnegative Ricci Curvature ◽  
The Green Function

Abstract Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci flow, Kähler–Ricci flow, and mean curvature flow, to name a few. As an elliptic analogue, Colding proved a sharp gradient estimate for the Green function on a manifold with nonnegative Ricci curvature. In this article, we prove a related matrix inequality on manifolds with suitable curvature and volume growth assumptions.

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