Critical Geller Equations on Complex Hyperbolic Space: Nondegeneracy, Geodesic Symmetry, and Global Compactness

Abstract

For each integer \(\ell\geq1\), the critical Geller equation on the Siegel domain \(\Ucal=\Heis^n\times(0,\infty)\) is the \(U(\ell)\)-invariant sector of the critical Folland--Stein equation on \(\Heis^{n+\ell}\) and, after a Cayley conjugation, a Brezis--Nirenberg equation on the rank-one symmetric space \(\CHyp^{n+1}\) in its harmonic \(AN\) realization \(\Heis^n\rtimes\R_+\); for \(\ell>1\) the conjugation also identifies the completed form domains. Working between the two descriptions, we obtain three main results. First, the bubble is nondegenerate in the invariant sector, and the exact normal Hessian-to-energy ratio is \(2/(n+\ell+4)\). Second, the optimal fixed-sector Bianchi--Egnell stability quotient lies strictly below both the local spectral threshold and the two-bubble threshold, and it is attained. Third, for an attractive Hardy perturbation at small coupling, the positive ground state on \(\CHyp^{n+1}\) is unique up to holomorphic isometry, geodesically radial, and nondegenerate modulo isometries; the symmetry follows from uniqueness in an equivariant normal slice, with no moving-plane argument. Along the way we determine the sharp Sobolev constant and all equality cases, classify nonnegative finite-energy solutions, prove a profile decomposition with no auxiliary concentration center, establish global Palais--Smale compactness with an exact action quantum, and compute the first normal deformation together with the second-order expansions of the sharp quotient and the action quantum. Finally, for a small repulsive potential, a Busemann barycenter--scale linking argument yields a positive solution with action strictly between one and two quanta.

60 pages, no figures