Vol. 246, No. 1, 2010

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The entropy formulas for the CR heat equation and their applications on pseudohermitian (2n + 1)-manifolds

Shu-Cheng Chang and Chin-Tung Wu

Vol. 246 (2010), No. 1, 1–29
Abstract

We derive Perelman’s and Nash-type entropy formulas for the CR heat equation on closed pseudohermitian (2n + 1)-manifolds and show that it is monotone nonincreasing if its pseudohermitian Ricci curvature minus (n + 1)2 times the pseudohermitian torsion is nonnegative. As results, we are able to obtain an integral version of the subgradient estimate for the CR heat equation and an upper bound estimate for the first positive eigenvalue of the sublaplacian by using the CR Bochner formulas and CR Harnack-type inequality. As a byproduct, we obtain a sharp lower bound estimate for the first positive eigenvalue of the sublaplacian on a closed pseudohermitian (2n + 1)-manifold.

Keywords
entropy formulas, sublaplacian, CR heat equation, CR Paneitz operator, gradient estimate, pseudohermitian manifold, pseudohermitian Ricci curvature, pseudohermitian torsion
Mathematical Subject Classification 2000
Primary: 32V05, 32V20
Secondary: 53C56
Milestones
Received: 10 February 2009
Revised: 10 January 2010
Accepted: 27 February 2010
Published: 1 May 2010
Authors
Shu-Cheng Chang
Department of Mathematics
National Taiwan University
Taipei 10617
Taiwan
Chin-Tung Wu
Department of Applied Mathematics
National Pingtung University of Education
Pingtung 90003
Taiwan