发表机构
University of Science and Technology of China（中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究完成了海森堡群上临界CR Yamabe方程正整体解的维度分类，通过利用Green表示的推论、共形协变性等，在无能量界的类中隔离出气泡流形，解决了欧氏方法无法适用的问题。
AI 中文摘要
我们证明，对于所有n≥2，海森堡群$\boldsymbol{\reals^n}$上的临界CR Yamabe方程$4\boldsymbol{\triangle_b} u = n^2 u^{(Q+2)/(Q-2)}$（其中$\boldsymbol{Q=2n+2}$）的每一个正整体解都是Jerison-Lee气泡。我们未对解的可积性、衰减性、有界性或对称性做任何假设。结合Catino、Li、Monticelli和Roncoroni在$\boldsymbol{\reals^1}$上的定理，这一结果完成了所有维度下正整体解的分类。欧氏空间中用于证明此类结论的两种方法在此均不适用：超平面反射不是CR自同构，CR反演仅在集合意义下保持其Koranyi球，因此解与其Kelvin变换的差在反演所在球上未必为零；也不存在可依赖的替代先验界——对每一个正解仅有的尺度不变估计是临界Morrey估计，而集中现象会使其饱和，既无法得到衰减性，也无法得到小质量正则性原理。我们转而从Green表示的两个精确推论出发，证明每一个正解都满足这两个推论：与气泡$\boldsymbol{U}$的互易性将到$\boldsymbol{U}$的距离转化为非负凸亏缺，无需线性二次型符号的信息；沿$\boldsymbol{U}$的共形轨道对同一互易性求导，可得到非线性重心恒等式，该恒等式禁止亏缺集中在CR球的单个点上。二者共同在无能量界的类中隔离出气泡流形，通过对该隔离量化Jerison-Lee张量亏缺，并将所得预算与Morrey估计交替使用，可将能量增长指数驱动至亏缺必须为零的范围。两个恒等式仅利用了共形协变性和精确的正Green表示。
英文摘要
We prove that, for every $n\ge 2$, 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 $\mathbb H^n$ 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.
Comments36 pages. v2: expanded discussion of previous results, added references and a note on independent work; no changes to the mathematics
