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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Li-Yau-Hamilton inequality for Yamabe flow on a closed CR $ 3$-manifold


Authors: Shu-Cheng Chang, Hung-Lin Chiu and Chin-Tung Wu
Journal: Trans. Amer. Math. Soc. 362 (2010), 1681-1698
MSC (2000): Primary 32V20; Secondary 53C44
Published electronically: November 17, 2009
MathSciNet review: 2574873
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Abstract: We deform the contact form by the (normalized) CR Yamabe flow on a closed spherical CR $ 3$-manifold. We show that if a contact form evolves with positive Tanaka-Webster curvature and vanishing torsion from initial data, then we obtain a new Li-Yau-Hamilton inequality for the CR Yamabe flow. By combining this parabolic subgradient estimate with a compactness theorem of a sequence of contact forms, it follows that the CR Yamabe flow exists for all time and converges smoothly to, up to the CR automorphism, a unique limit contact form of positive constant Webster scalar curvature on a closed CR $ 3$-manifold, which is CR equivalent to the standard CR $ 3$-sphere with positive Tanaka-Webster curvature and vanishing torsion.


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Additional Information

Shu-Cheng Chang
Affiliation: Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan, Republic of China
Email: scchang@math.ntu.edu.tw

Hung-Lin Chiu
Affiliation: Department of Mathematics, National Central University, Chung-Li 32054, Taiwan, Republic of China
Email: hlchiu@math.ncu.edu.tw

Chin-Tung Wu
Affiliation: Department of Applied Mathematics, National PingTung University of Education, PingTung 90003, Taiwan, Republic of China
Email: ctwu@mail.npue.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9947-09-05011-9
PII: S 0002-9947(09)05011-9
Keywords: Li-Yau-Hamilton inequality, CR Bochner formula, Tanaka-Webster curvature, pseudoharmonic manifold, CR pluriharmonic operator, CR Paneitz operator, sub-Laplacian, subgradient estimate, CR Yamabe flow, positive mass theorem.
Received by editor(s): January 23, 2007
Published electronically: November 17, 2009
Additional Notes: This research was supported in part by the NSC of Taiwan
Article copyright: © Copyright 2009 American Mathematical Society