Critical Geller Equations on Complex Hyperbolic Space: Sharp Stability, Ground-State Symmetry, and Global Compactness

Abstract

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.