Classification of positive entire solutions of the CR Yamabe equation on the Heisenberg group
Abstract
We prove that, for every n≥ 2, every positive entire solution of the critical CR Yamabe equation 4Δb u = n2 u(Q+2)/(Q−2), Q=2n+2, on the Heisenberg group ℍn is a Jerison-Lee bubble. No integrability, decay, boundedness, or symmetry is assumed. Together with the theorem of Catino, Li, Monticelli, and Roncoroni in ℍ1, 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.
Related
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BibTeX
@misc{li2026classification,
title = {{Classification of positive entire solutions of the CR Yamabe equation on the Heisenberg group}},
author = {Jungang Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.10642},
eprint = {2608.10642},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.10642},
}
RIS
TY - UNPB
TI - Classification of positive entire solutions of the CR Yamabe equation on the Heisenberg group
AU - Jungang Li
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.10642
UR - https://arxiv.org/abs/2608.10642
AB - We prove that, for every n≥ 2, every positive entire solution of the critical CR Yamabe equation 4Δ_b u = n^2 u^(Q+2)/(Q−2), Q=2n+2, on the Heisenberg group 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 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.
ER -