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CR Yamabe Equation on the Heisenberg Group via the method of moving spheres

Congwen Liu

TL;DR

The paper resolves the global Liouville classification for positive solutions of the CR Yamabe equation on the Heisenberg group by extending the method of moving spheres to the CR setting. It introduces a CR-adapted inversion framework, establishes Li–Nirenberg–Zhu type calculus lemmas in $\mathbb{H}^n$, and proves a Terracini-type integral inequality to replace the maximum principle. These tools yield that every positive solution must be a Jerison–Lee bubble, with explicit bubble parameters, removing prior finite-energy or symmetry restrictions. The results provide a complete three- to five-sentence overview of the CR Yamabe problem on $\mathbb{H}^n$ and its sharp extremals, mirroring classical Euclidean Yamabe theory in a noncommutative, sub-Riemannian context.

Abstract

In this paper, we classify positive solutions to the CR Yamabe equation on the Heisenberg group $\mathbb{H}^n$. We show that all such solutions are Jerison-Lee bubbles, without imposing any finite-energy or a priori symmetry assumptions. This result can be regarded as an analogue for $\mathbb{H}^n$ of the celebrated Caffarelli-Gidas-Spruck classification theorem in $\mathbb{R}^n$. To establish this, we develop a systematic approach to implement the method of moving spheres in the setting of the Heisenberg group.

CR Yamabe Equation on the Heisenberg Group via the method of moving spheres

TL;DR

The paper resolves the global Liouville classification for positive solutions of the CR Yamabe equation on the Heisenberg group by extending the method of moving spheres to the CR setting. It introduces a CR-adapted inversion framework, establishes Li–Nirenberg–Zhu type calculus lemmas in , and proves a Terracini-type integral inequality to replace the maximum principle. These tools yield that every positive solution must be a Jerison–Lee bubble, with explicit bubble parameters, removing prior finite-energy or symmetry restrictions. The results provide a complete three- to five-sentence overview of the CR Yamabe problem on and its sharp extremals, mirroring classical Euclidean Yamabe theory in a noncommutative, sub-Riemannian context.

Abstract

In this paper, we classify positive solutions to the CR Yamabe equation on the Heisenberg group . We show that all such solutions are Jerison-Lee bubbles, without imposing any finite-energy or a priori symmetry assumptions. This result can be regarded as an analogue for of the celebrated Caffarelli-Gidas-Spruck classification theorem in . To establish this, we develop a systematic approach to implement the method of moving spheres in the setting of the Heisenberg group.
Paper Structure (12 sections, 14 theorems, 168 equations)

This paper contains 12 sections, 14 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Heisenberg Group
  4. CR maps and the generalized Kelvin transform on $\mathbb{H}^n$
  5. Basic properties of the generalized CR inversion $\Phi_{\xi, \lambda}^{\, \beta}$
  6. Proof of Proposition \ref{['prop:justifydefn']}
  7. Proof of Theorem \ref{['thm:lnzlem1']}
  8. Preparations for Proof of Theorem \ref{['thm:lnzlem2']}
  9. Proof of Theorem \ref{['thm:lnzlem2']}
  10. Terracini-type integral inequality
  11. The Proof of Theorem \ref{['thm:main']}
  12. An alternative proof of Theorem D

Key Result

Theorem 1.1

For $n \geq 1$, every positive solution $u$ to eqn:cr_yamabe is a Jerison–Lee bubble, that is, for some $K>0$, $\mu\in\mathbb{C}^n$, $\kappa\in\mathbb{C}$ with $\operatorname{Im}\kappa > \frac{|\mu|^2}{4}$.

Theorems & Definitions (30)

  • Theorem 1.1: Main Theorem
  • Definition 1.2
  • Proposition 1.3
  • Theorem 1.4: Calculus Lemma I of Li-Nirenberg-Zhu type
  • Theorem 1.5: Calculus Lemma II of Li-Nirenberg-Zhu type
  • Definition 2.1: Generalized Kelvin Transform
  • Lemma 2.2
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • ...and 20 more