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

Chang, Shu-Cheng; Chiu, Hung-Lin; Wu, Chin-Tung

doi:10.1090/S0002-9947-09-05011-9

The Li-Yau-Hamilton inequality for Yamabe flow on a closed CR $3$-manifold
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Trans. Amer. Math. Soc. 362 (2010), 1681-1698 Request permission

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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