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

TL;DR

This paper proves that the CR Q-curvature flow on certain closed strictly pseudoconvex CR 3-manifolds exists globally and converges smoothly, affirming a long-standing conjecture about the existence of contact forms with zero CR Q-curvature.

Contribution

It establishes the global existence and convergence of the CR Q-curvature flow on embeddable CR 3-manifolds with specific geometric conditions, confirming the conjecture in these cases.

Findings

Flow exists for all time

Flow converges smoothly to a contact form with zero CR Q-curvature

Affirms the conjecture for manifolds with vanishing first Chern class and torsion

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 $\theta $ 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 $(M,\ J,[\theta_{0}])$ of the vanishing first Chern class $c_{1}(T_{1,0}M)$. Suppose that $M$ is embeddable and the CR Paneitz operator $P_{0}$ 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 $M\times \lbrack 0,\infty ).$\ 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.