IJPAM: Volume 93, No. 3 (2014)

GRADIENT ESTIMATES FOR HEAT-TYPE EQUATIONS
ON MANIFOLDS EVOLVING BY THE RICCI FLOW

Abimbola Abolarinwa
School of Mathematical and Physical Sciences
University of Sussex
Brighton, BN1 9QH, UK


Abstract. In this paper, certain localized and global gradient estimates for all positive solutions to the geometric heat equation coupled to the Ricci flow either forward or backward in time are proved. As a by product, we obtain various Li-Yau type differential Harnack estimates. We also discuss the case when the diffusion operator is perturbed with the curvature operator (precisely, when the Laplacian is replaced with " $ \Delta - R(x,t)$", $R$ being the scalar operator). This is well generalised to the case of an adjoint heat equation under the Ricci flow.

Received: March 17, 2014

AMS Subject Classification: 35K05, 53C25, 53C44

Key Words and Phrases: Ricci flow, conjugate heat equation, Harnack inequalities, gradient estimates, Laplace-Beltrami operator, Laplacian comparison theorem

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DOI: 10.12732/ijpam.v93i3.14 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 93
Issue: 3
Pages: 463 - 489

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).