Results Related to Prescribing Pseudo-Hermitian Scalar Curvature

Abstract

In this paper, we consider the problem of prescribing pseudo-Hermitian scalar curvature on a compact strictly pseudoconvex CR manifold M. Using geometric flow, we prove that for any negative smooth function f we can prescribe the pseudo-Hermitian scalar curvature to be f, provided that dim M = 3 and the CR Yamabe invariant of M is negative. On the other hand, we establish some uniqueness and non-uniqueness results on prescribing pseudo-Hermitian scalar curvature.

The Webster scalar curvature flow on CR sphere. Part I

“…The following lemma is proved in [27] which follows from differentiating (2.6). We refer the reader to [27] for the details of the proof.…”

“…For the case R θ 0 > 0, Chang, Chiu, and Wu [8] proved the convergence when M is spherical and n = 1 and θ 0 is torsion-free. For general n, the author proved in [27] the long time existence for the case when R θ 0 > 0, and the convergence when M is the CR sphere. As a generalization of CR Yamabe problem, one can consider prescribing CR Webster scalar curvature problem: given a function f on a CR manifold (M, θ 0 ), we want to find a contact form θ conformal to θ 0 such that its Webster scalar curvature R θ = f .…”

“…As a generalization of CR Yamabe problem, one can consider prescribing CR Webster scalar curvature problem: given a function f on a CR manifold (M, θ 0 ), we want to find a contact form θ conformal to θ 0 such that its Webster scalar curvature R θ = f . This has been studied in [17,19,23,27,[36][37][38]. As an analogy of Nirenberg's problem, we study the problem of prescribing Webster scalar curvature on the CR sphere (S 2n+1 , θ 0 ).…”

The Webster scalar curvature flow on CR sphere. Part II

“…We are interested in the following question: can we find a contact form θ conformal to θ 0 such that its Webster scalar curvature R θ = f ? This has been studied in [9,11,14,16,23,24,25]. When f is constant, this is the CR Yamabe problem, which was solved by Jerison and Lee in [18,19,20], and by Gamara and Yacoub in [13,15].…”

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

“…As an analogy of the Nirenberg's problem, we ask the following question: can we find a contact form θ conformal to θ 0 such that its Webster scalar curvature R θ = f ? 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: …”