Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Compactness in Dimension Five and Equivariant Noncompactness for the CR Yamabe Problem

Claudio Afeltra, Andrea Pinamonti, Pak Tung Ho

Abstract

We study compactness and noncompactness phenomena for the CR Yamabe equation on compact strictly pseudoconvex CR manifolds. First, in dimension five we establish uniform \emph{a priori} estimates for families of positive solutions of subcritical equations for the conformal CR sub-Laplacian \[ L_{J}u = u^{p}, \] with $p$ bounded away from the critical exponent, assuming positivity of the CR Yamabe constant and positivity of the $p$-mass at every point. As a consequence, the corresponding set of solutions is precompact in Hölder topologies. Secondly, we consider the equivariant CR Yamabe problem for a compact subgroup $G$ of pseudo-Hermitian transformations. We construct a $G$-invariant CR structure on $S^{3}$, not equivalent to the standard one, for which the associated CR Yamabe equation admits a sequence of $G$-invariant solutions whose maxima diverge, thereby proving noncompactness in the equivariant setting. The arguments combine a Pohozaev-type identity in pseudohermitian normal coordinates with a blow-up analysis and Liouville-type classification results on the Heisenberg group.

Compactness in Dimension Five and Equivariant Noncompactness for the CR Yamabe Problem

Abstract

We study compactness and noncompactness phenomena for the CR Yamabe equation on compact strictly pseudoconvex CR manifolds. First, in dimension five we establish uniform \emph{a priori} estimates for families of positive solutions of subcritical equations for the conformal CR sub-Laplacian with bounded away from the critical exponent, assuming positivity of the CR Yamabe constant and positivity of the -mass at every point. As a consequence, the corresponding set of solutions is precompact in Hölder topologies. Secondly, we consider the equivariant CR Yamabe problem for a compact subgroup of pseudo-Hermitian transformations. We construct a -invariant CR structure on , not equivalent to the standard one, for which the associated CR Yamabe equation admits a sequence of -invariant solutions whose maxima diverge, thereby proving noncompactness in the equivariant setting. The arguments combine a Pohozaev-type identity in pseudohermitian normal coordinates with a blow-up analysis and Liouville-type classification results on the Heisenberg group.
Paper Structure (5 sections, 34 theorems, 171 equations)

This paper contains 5 sections, 34 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm2']}
  4. Compactness
  5. Blow-up analysis

Key Result

Theorem 1.1

Let $(M,J,\theta)$ be a compact $3$-dimensional strictly pseudoconvex CR manifold of positive CR Yamabe constant such that, for every $x\in M$, its $p$-mass at $x$ is positive, i.e. $m_x>0$. Then, for every $\epsilon>0$ and $k\in\mathbb{N}$, there exists a constant $C$ such that for every $u\in \cup_{1+\epsilon\leq p\leq 3}\mathcal{M}_p$ and $0<\alpha<1$. Here where $\Gamma_{k,\alpha}$ is the Hö

Theorems & Definitions (62)

  • Theorem 1.1: Theorem 1.1 in Afeltra
  • Corollary 1.2: Corollary 1.2 in Afeltra
  • Theorem 1.3
  • Conjecture 1.4: Equivariant CR Yamabe problem
  • Theorem 1.5: Theorem 1.1 in AfeltraPinamonti
  • Theorem 1.6
  • Theorem 2.1: Lemma 5 from MalchiodiUguzzoni
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • ...and 52 more