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

TL;DR

This paper classifies all positive solutions to the CR Yamabe equation on the Heisenberg group, showing they are Jerison-Lee bubbles, by developing a systematic moving spheres method without energy or symmetry assumptions.

Contribution

It introduces a systematic approach to apply the method of moving spheres in the Heisenberg group setting, leading to a complete classification of solutions.

Findings

All positive solutions are Jerison-Lee bubbles

No finite-energy or symmetry assumptions needed

Method extends the moving spheres technique to sub-Riemannian geometry

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.