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

TL;DR

This paper establishes uniform a priori estimates for solutions to the CR Yamabe equation in dimension five and constructs examples demonstrating noncompactness in the equivariant setting, highlighting differences in solution behavior.

Contribution

It provides the first uniform estimates for subcritical CR Yamabe solutions in dimension five and constructs noncompactness examples under symmetry constraints.

Findings

Uniform a priori estimates for solutions in dimension five

Existence of noncompactness in the equivariant CR Yamabe problem

Construction of G-invariant structures with diverging solutions

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 H\"older 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.