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A compactness result for the CR Yamabe problem in three dimensions

Claudio Afeltra

Abstract

We prove the compactness of the set of solutions to the CR Yamabe problem on a compact strictly pseudoconvex CR manifold of dimension three whose blow-up manifolds at every point have positive p-mass. As a corollary we deduce that compactness holds for CR-embeddable manifolds which are not CR-equivalent to $S^3$. The theorem is proved by blow-up analysis.

A compactness result for the CR Yamabe problem in three dimensions

Abstract

We prove the compactness of the set of solutions to the CR Yamabe problem on a compact strictly pseudoconvex CR manifold of dimension three whose blow-up manifolds at every point have positive p-mass. As a corollary we deduce that compactness holds for CR-embeddable manifolds which are not CR-equivalent to . The theorem is proved by blow-up analysis.
Paper Structure (8 sections, 27 theorems, 238 equations)

This paper contains 8 sections, 27 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. A review of CR geometry
  4. The Heisenberg group
  5. Function spaces and estimates for the sublaplacian
  6. A Pohozaev identity
  7. Blow-up analysis
  8. Some computations in pseudohermitian normal coordinates

Key Result

Theorem 1.1

Let $(M,\mathscr{H},\theta)$ be a three-dimensional strictly pseudoconvex pseudohermitian manifold of positive CR Yamabe class such that, for every $x\in M$, $m_x>0$. Then for every $\varepsilon>0$ and $k\in\mathbf{N}$ there exists a constant $C$ such that for every $u\in\cup_{1+\varepsilon\le p\le 3}\mathscr{M}_p$ and $0<\alpha<1$, where $L_{\theta}=-4\Delta_b+R$ being the conformal sublaplacia

Theorems & Definitions (55)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 45 more