Global existence and convergence for the CR Q-curvature flow in a closed strictly pseudoconvex CR 3-manifold

2019/05/06 by Shu-Cheng Chang, Chang, Shu-Cheng, Ting-Jung Kuo +3
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory #math.CV #math.DG #msc:32V20 #msc:53C44

paper · pdf · doi:10.48550/arxiv.1905.01783

30 pages. arXiv admin note: text overlap with arXiv:math/0510494

arxiv created 2019/07/05 · arxiv updated 2019/07/08

Abstract

In this note, we affirm the partial answer to the long open Conjecture which states that any closed embeddable strictly pseudoconvex CR 3-manifold admits a contact form θ with the vanishing CR Q-curvature. More precisely, we deform the contact form according to an CR analogue of Q%-curvature flow in a closed strictly pseudoconvex CR 3-manifold (M, J,[θ₀]) of the vanishing first Chern class c₁(T_1,0M). Suppose that M is embeddable and the CR Paneitz operator P₀ is nonnegative with kernel consisting of the CR pluriharmonic functions. We show that the solution of CR Q-curvature flow exists for all time and has smoothly asymptotic convergence on M× \lbrack 0,∞ ). As a consequence, we are able to affirm the Conjecture in a closed strictly pseudoconvex CR 3-manifold of the vanishing first Chern class and vanishing torsion.

Citations

• A Fourth Order Curvature Flow on a CR 3-manifold

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