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.

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 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.

scholarly journals Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds

2020 ◽  
Vol 2020 (764) ◽  
pp. 71-109 ◽  
Author(s):  
Paul Bryan ◽  
Mohammad N. Ivaki ◽  
Julian Scheuer
Keyword(s):  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Mean Curvature Flow ◽  
Einstein Manifolds ◽  
De Sitter ◽  
Harnack Estimate ◽  
Nonnegative Sectional Curvature ◽  
Harnack Inequalities ◽  
Harnack Estimates ◽  
New Type

AbstractWe obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant nonnegative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifolds of nonnegative sectional curvature. Using a concept of “duality” for strictly convex hypersurfaces, we also obtain a new type of inequality, so-called “pseudo”-Harnack inequality, for expanding flows in the sphere and in the hyperbolic space.

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.

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