scholarly journals The conjugate heat equation and Ancient solutions of the Ricci flow

2011 ◽  
Vol 228 (5) ◽  
pp. 2891-2919 ◽  
Author(s):  
Xiaodong Cao ◽  
Qi S. Zhang
Keyword(s):  
Heat Equation ◽  
Ricci Flow ◽  
Ancient Solutions ◽  
Conjugate Heat Equation

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 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.

scholarly journals Ancient solutions for Andrews’ hypersurface flow

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Peng Lu ◽  
Jiuru Zhou
Keyword(s):  
Ricci Flow ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Euclidean Spaces ◽  
Round Cylinder ◽  
Ancient Solutions ◽  
Singular Time

AbstractWe construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions {S^{J}\times\mathbb{R}^{n-J}}, {1\leq J\leq n-1}. These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.

scholarly journals Ancient solutions of the Ricci flow on bundles

2017 ◽  
Vol 318 ◽  
pp. 411-456 ◽  
Author(s):  
Peng Lu ◽  
Y.K. Wang
Keyword(s):  
Ricci Flow ◽  
Ancient Solutions
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