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Heat Kernel Estimates Under the Ricci–Harmonic Map Flow

Part of: Parabolic equations and systems Global differential geometry

Published online by Cambridge University Press:  22 February 2017

Mihai Băileşteanu  and
Hung Tran
Mihai Băileşteanu
Affiliation:
Department of Mathematics, Central Connecticut State University, 120 Marcus White Hall, New Britain, CT 06052, USA (mihaib@ccsu.edu)
Hung Tran
Affiliation:
Department of Mathematics, University of California, Irvine, 440G Rowland Hall, Irvine, CA 92612, USA (hungtt1@uci.edu)
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Abstract

This paper considers the Ricci flow coupled with the harmonic map flow between two manifolds. We derive estimates for the fundamental solution of the corresponding conjugate heat equation and we prove an analogue of Perelman's differential Harnack inequality. As an application, we find a connection between the entropy functional and the best constant in the Sobolev embedding theorem in ℝn .

Keywords

heat equationheat kernelRicci flowharmonic mapSobolev embeddingdifferential Harnack inequality

MSC classification

Secondary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 53C43: Differential geometric aspects of harmonic maps 35K55: Nonlinear parabolic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 4 , November 2017 , pp. 831 - 857
Copyright
Copyright © Edinburgh Mathematical Society 2017 

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