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

Abstract

We deform the contact form by the (normalized) CR Yamabe flow on a closed spherical CR 3 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 3 -manifold, which is CR equivalent to the standard CR 3 3 -sphere with positive Tanaka-Webster curvature and vanishing torsion.