A CR structure with blowing up solutions to the CR Yamabe problem

TL;DR

This paper constructs a specific CR structure on the 3-sphere where solutions to the CR Yamabe problem are non-compact and exhibit blow-up behavior, using deformation techniques and the Lyapunov-Schmidt method.

Contribution

It demonstrates the existence of a CR structure on S^3 with non-compact solution sets to the CR Yamabe problem, introducing a new deformation approach and analytical proof.

Findings

Existence of a non-compact solution set to the CR Yamabe problem on a deformed CR structure.

Construction of a CR structure on S^3 with blowing-up solutions.

Application of the Lyapunov-Schmidt method to prove solution blow-up.

Abstract

We prove the existence of a CR structure on $S^3$ such that the set of solutions to the CR Yamabe problem is not compact and admits a blowing-up sequence. Such CR structure is built deforming the standard CR structure of $S^3$ in the direction of the Rossi sphere CR structure on small balls, and the existence of the blowing-up sequence of solutions is proved through the Lyapunov-Schmidt method.