The Long-Time Existence and Convergence of the Cr Yamabe Flow

Abstract

In this paper, we prove the long-time existence of the CR Yamabe flow on the compact strictly pseudoconvex CR manifold with positive CR invariant. We also prove the convergence of the CR Yamabe flow on the sphere by proving that: the contact form which is pointwise conformal to the standard contact form on the sphere converges exponentially to a contact form of constant pseudo-Hermitian sectional curvature. We also show that the eigenvalues of some geometric operators are non-decreasing under the unnormalized CR Yamabe flow provided that the pseudo-Hermitian scalar curvature satisfies certain conditions.

The Webster scalar curvature flow on CR sphere. Part I

“…See Theorem 5.3. In particular, this recovers the result of the author in [28]. In Appendix A, we prove the CR analogue of the Aubin's improvement of the Sobolev inequality.…”

“…Our aim is to prove the following result, which recovers the result of the author in [28]: Proof. First we show the exponential decay of F 2 .…”

Prescribed Webster scalar curvature on S2+1 in the presence of reflection or rotation symmetry

“…This has been studied in [9,13,14,16,23,25,26]. Using the Webster scalar curvature flow (see (2.1) below) which is a generalization of the CR Yamabe flow (see [15]), we have proved in [18] and [19] the following: …”

Convergence of the CR Yamabe flow

“…For the case Y (M, θ 0 ) > 0, Chang-Chiu-Wu [8] proved the convergence of the CR Yamabe flow when M is spherical and n = 1 and θ 0 is torsion-free. For general n, the long time existence was proved by the first author in [24] for the case when Y (M, θ 0 ) > 0. In [24], the first author also proved the convergence of the CR Yamabe flow when M is the CR sphere (see [25] for an alternative proof).…”

“…For general n, the long time existence was proved by the first author in [24] for the case when Y (M, θ 0 ) > 0. In [24], the first author also proved the convergence of the CR Yamabe flow when M is the CR sphere (see [25] for an alternative proof).…”