Classification of positive entire solutions of the CR Yamabe equation on the Heisenberg group

Abstract

We prove that, for every $n\ge 2$, every positive entire solution of the critical CR Yamabe equation $4\Delta_b u = n^2 u^{(Q+2)/(Q-2)}$, $Q=2n+2$, on the Heisenberg group $\mathbb H^n$ is a Jerison-Lee bubble. No integrability, decay, boundedness, or symmetry is assumed. Together with the theorem of Catino, Li, Monticelli, and Roncoroni in $\mathbb H^1$, this classifies the positive entire solutions in every dimension. Both Euclidean routes to such a statement lose their starting configuration here. Hyperplane reflections are not CR automorphisms, and a CR inversion preserves its Koranyi sphere only setwise, so the difference between a solution and its Kelvin transform need not vanish on the sphere one inverts in. Nor is there a substitute a priori bound to fall back on: the only scale-invariant estimate available for every positive solution is a critical Morrey bound, which concentration saturates and which yields neither decay nor a small-mass regularity principle. We proceed instead from two exact consequences of the Green representation, which every positive solution is shown to satisfy. Reciprocity with a bubble $U$ converts the distance from $U$ into a nonnegative convex deficit, so that no information about the sign of a linearized quadratic form is required; differentiating the same reciprocity along the conformal orbit of $U$ gives a nonlinear barycentre identity, which forbids that deficit from concentrating at a single point of the CR sphere. Together these isolate the bubble manifold in a class carrying no energy bound. Quantizing the Jerison-Lee tensor defect against this isolation, and alternating the resulting budget with the Morrey bound, drives the energy-growth exponent into the range where the defect must vanish. Both identities use only conformal covariance and an exact positive Green representation.