The Webster scalar curvature flow on CR sphere. Part I

Abstract

This is the first of two papers, in which we prove some properties of the Webster scalar curvature flow. More precisely, we establish the long-time existence, L^p convergence and the blow-up analysis for the solution of the flow. As a by-product, we prove the convergence of the CR Yamabe flow on the CR sphere. The results in this paper will be used to prove a result of prescribing Webster scalar curvature on the CR sphere, which is the main result of the second paper.Comment: To appear in Advances in Mathematic

The Webster scalar curvature flow on CR sphere. Part II

“…By (2.9) and Lemma 2.3, and by (3.4) in [17], the first term on the right hand side of (4.7) can be bounded by…”

“…By the long existence of the flow (2.4) which was proved in part I, we know that u i (t), i = 1, 2 are smooth in any given finite time interval [0, T ]. Moreover, by Lemma 2.8 in [17], there exists constant…”

“…It follows from Theorem 2.2 that we have the following dichotomy: Either the flow converges in S p 2 for some p > 2n + 2, and in this case, f can be realized as the Webster scalar curvature of some contact form conformal to θ 0 thanks to Lemma 4.8 in [17], or the corresponding normalized flow h(t) defined in (2.10) converges.…”

“…In [17], we established the long-time existence of the flow (2.2). See section 2.3 in [17]. Define (2.6)…”

“…We introduced the Webster scalar curvature flow in part I [17], which is defined as the evolution of the contact form θ = θ(t), t ≥ 0 as follows:…”