AI 中文总结
本文研究复双曲空间上的临界Geller方程，确定其Sobolev常数、解的分类等性质，构造了相关正解并证明了全局紧性等结果。
AI 中文摘要
对于每个整数ℓ≥1，Siegel域𝒰=ℋⁿ×(0,∞)上的临界Geller方程是ℋⁿ⁺¹上的临界Folland–Stein方程的U(ℓ)-不变子域，其光滑方程与复双曲空间ℂℍⁿ⁺¹上的Brezis–Nirenberg方程恰为Cayley共轭关系。当ℓ>1时，该共轭还可识别完备形式域。我们确定了最佳Sobolev常数及所有等号情形，对非负有限能量解进行分类，证明了无辅助集中中心的剖面分解，并建立了Bianchi–Egnell稳定性不等式。其最优固定子域商严格低于局部和双气泡阈值且可达；该气泡在不变子域中非退化，精确的法Hessian-能量比为2/(n+ℓ+4)。对于ℓ>1且在 coercive范围0<λ<(ℓ-1)²内的吸引Hardy扰动，我们构造了光滑正最小作用解并证明具有精确作用量子的全局Palais–Smale紧性。在足够小的耦合下，正归一化极小值在全纯等距下形成单一轨道，每个正内在极小值均为测地径向，基态在全纯等距模下非退化，该对称性由等变法向切片唯一性导出，无需移动平面论证。我们还确定了第一法向变形及最佳商与作用量子的二阶展开。最后，对于小的排斥势，Busemann重心-尺度连接论证给出了作用严格介于1和2个量子之间的正解。
英文摘要
For integers \(n,\ell\geq1\), set \(Q_\ell=2(n+\ell)+2\) and \(q_\ell=2Q_\ell/(Q_\ell-2)\). On the Siegel domain \(\Ucal=\Heis^n\times(0,\infty)\), we study \[ -Δ_{\Heis^n}v -4ρ\bigl(v_{ρρ}+T^2v\bigr)-4\ell v_ρ =|v|^{q_\ell-2}v. \] For its Dirichlet form \(E_\ell\), we determine the sharp Sobolev constant and all extremals, prove \[ S_\ell\|v\|_{q_\ell}^2\leq E_\ell(v),\qquad E_\ell(v)-S_\ell\|v\|_{q_\ell}^2 \geqκ_{n,\ell} \operatorname{dist}_{\dot S^1_\ell} \bigl(v,\mathfrak M_\ell^{\R}\bigr)^2, \] where \(\mathfrak M_\ell^{\R}\) is the extremal cone. We classify nonnegative finite-energy solutions and establish the linearized kernel, profile decomposition, and attainment of the optimal stability quotient. Cayley conjugation yields ground-state symmetry and nondegeneracy, global Palais--Smale compactness, and perturbative existence for critical equations on \(\CHyp^{n+1}\). The mechanism is the radial lift \(v^\uparrow(z,w,t)=v(z,t,|w|^2)\), which reverses the interior-to-boundary construction by realizing the Siegel domain as a symmetry-reduced slice of a larger Heisenberg group. The completed-space reduction resolves the degenerate axis, hidden auxiliary concentration, and the mismatch of extremal cones. Together with the Cayley transform, this supplies the missing nonlinear layer---classification, stability, bubbling, and variational compactness---on complex hyperbolic space and provides a blueprint for other rank-one symmetric spaces.
Comments60 pages, no figures
