scholarly journals Gradient estimates for the heat equation under the Ricci flow

2010 ◽  
Vol 258 (10) ◽  
pp. 3517-3542 ◽  
Author(s):  
Mihai Bailesteanu ◽  
Xiaodong Cao ◽  
Artem Pulemotov
Keyword(s):  
Heat Equation ◽  
Ricci Flow ◽  
Gradient Estimates

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.

GRADIENT ESTIMATES VIA TWO-POINT FUNCTIONS FOR PARABOLIC EQUATIONS UNDER RICCI FLOW

2020 ◽  
Vol 102 (2) ◽  
pp. 319-330
Author(s):  
MIN CHEN
Keyword(s):  
Riemannian Manifolds ◽  
Ricci Flow ◽  
Parabolic Equations ◽  
Gradient Estimates ◽  
Quasilinear Parabolic Equations ◽  
Quasilinear Parabolic

We derive estimates relating the values of a solution at any two points to the distance between the points for quasilinear parabolic equations on compact Riemannian manifolds under Ricci flow.

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