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

Claudio Afeltra, Andrea Pinamonti

TL;DR

The paper constructs a CR structure on $S^3$ with non-compact CR Yamabe solution set by deforming the standard structure toward Rossi spheres on small balls and applying a Lyapunov–Schmidt reduction to produce a blowing-up sequence of solutions. This mirrors Brendle’s blow-up strategy in the Riemannian Yamabe problem and leverages the Rossi deformation, which exhibits negative pseudohermitian mass, to break compactness in dimension three. The authors develop a finite-dimensional reduction around bubble solutions on the Heisenberg group, establish precise asymptotics for the perturbed CR Yamabe functional on Rossi-like bubbles, and patch local deformations to build a global counterexample on $S^3$. The result clarifies the role of pseudohermitian mass in compactness phenomena and provides a robust CR-analytic framework for constructing blow-up sequences in CR geometry.

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.

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

TL;DR

The paper constructs a CR structure on with non-compact CR Yamabe solution set by deforming the standard structure toward Rossi spheres on small balls and applying a Lyapunov–Schmidt reduction to produce a blowing-up sequence of solutions. This mirrors Brendle’s blow-up strategy in the Riemannian Yamabe problem and leverages the Rossi deformation, which exhibits negative pseudohermitian mass, to break compactness in dimension three. The authors develop a finite-dimensional reduction around bubble solutions on the Heisenberg group, establish precise asymptotics for the perturbed CR Yamabe functional on Rossi-like bubbles, and patch local deformations to build a global counterexample on . The result clarifies the role of pseudohermitian mass in compactness phenomena and provides a robust CR-analytic framework for constructing blow-up sequences in CR geometry.

Abstract

We prove the existence of a CR structure on 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 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.
Paper Structure (6 sections, 13 theorems, 157 equations)

This paper contains 6 sections, 13 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rossi spheres
  4. Lyapunov-Schmidt method
  5. Construction of CR structures with large solutions
  6. Estimates for the sublaplacian and Webster curvature

Key Result

Theorem 1.1

There exists a CR structure on $S^3$, not equivalent to the standard one, such that the associated CR Yamabe equation (where $L_{J,\theta}$ is the conformal sublaplacian, defined in Section SezionePreliminari) has a set of solutions $\{u_k\}_{k\in\mathbf{N}}$ with $\max u_k\to\infty$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • proof
  • ...and 10 more