Hausdorff dimension of limit sets for spherical CR manifolds
Let \(M^{2n+1} (n\geq 1)\) be a compact, \((2n+1)\)-dimensional, strictly pseudoconvex \(CR\)-manifold. \(M^{2n+1}\) is spherical if it is everywhere locally \(CR\)-equivalent to the unit sphere \(S^{2n+1} \subset \mathbb{C}^{n+1}\). If \(\widetilde M^{2n+1}\) is the universal cover of \(M^{2n+1}\), then the \(CR\)-developing map \(\Phi: \widetilde M^{2n+1} \to S^{2n+1}\) is a \(CR\)-immersion and \(M^{2n+1} = \Omega/ \Gamma\) where \(\Omega\) is a simply connected open set in \(S^{2n+1}\) and \(\Gamma\) is a complex Klein group acting on \(\Omega\) properly discontinuously. The author makes the following assumptions on the spherical \(CR\)-manifold \(M^{2n+1}\): (1) the \(CR\)-developing map is injective, (2) the \(CR\)-Yamabe invariant \(\lambda (M^{2n+1})\) is positive. The main result is given by Theorem 2. Under assumptions 1 and 2, the Carnot-Hausdorff dimension of the limit set of the fundamental group \(\pi_1 (M^{2n+1})\) is bounded above by the number \(n\).\(s(M^{2n+1})\), where \(s(M^{2n+1})\) is a \(CR\)-invariant satisfying \(s(M^{2n+1}) \leq 1\).
scientific article; zbMATH DE number 944389