The Webster scalar curvature flow on CR sphere. Part I

TL;DR

This paper studies the Webster scalar curvature flow on the CR sphere, proving long-term existence, convergence, and blow-up behavior, and establishing the convergence of the CR Yamabe flow, with applications to prescribing Webster scalar curvature.

Contribution

It provides the first detailed analysis of the Webster scalar curvature flow on the CR sphere, including long-time existence and convergence results, laying groundwork for prescribing curvature problems.

Findings

Proved long-time existence of the flow

Established L^p convergence of the flow

Analyzed blow-up behavior of solutions

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.