Blow-up analysis and degree theory for the Webster curvature prescription problem in three dimensions

TL;DR

This paper studies the Webster curvature prescription problem on three-dimensional CR manifolds, establishing solution compactness, computing the Leray-Schauder degree, and deriving new existence results under mild hypotheses.

Contribution

It introduces a blow-up analysis and degree theory approach to the Webster curvature problem, extending previous results to broader conditions.

Findings

Solutions form a compact set under given conditions

Leray-Schauder degree is explicitly computed

New existence results are established for the problem

Abstract

Given a strictly pseudoconvex CR manifold $M$ of dimension three and positive CR Yamabe class, and a positive smooth function $K:M\to\mathbf{R}$ verifying some mild and generic hypotheses, we prove the compactness of the set of solutions of the Webster curvature prescription problem associated to $K$, and we compute the Leray-Schauder degree in terms of the critical points of $K$. As a corollary, we get an existence result which generalizes the ones existent in the literature.