A generalization of the Greene-Krantz theorem for the semicontinuity property of automorphism groups
The author proves the following generalisation for CR-manifolds of the Greene-Krantz theorem [\textit{R. E. Greene} and \textit{S. G. Krantz}, Math. Ann. 261, 425--446 (1982; Zbl 0531.32016)] on the semicontinuity of the automorphism groups of a convergent sequence of bounded strongly pseudoconvex domains: Let \(\{J_k: k = 1, 2, \dots \}\) be a sequence of \(C^\infty\)-smooth strongly pseudoconvex CR structures on a compact differentiable manifold \(M\) of dimension \(2n + 1\) which converges to a \(C^\infty\)-smooth strongly pseudoconvex CR structure \(J_0\) on \(M\) in the \(C^\infty\) sense. Suppose that the CR automorphism group \(\text{Aut}_{\mathrm{CR}}(M, J_0)\) is compact. Then there exists \(N > 0\) and a diffeomorphism \(\phi_k: M \to M\) for each \(k > N\) such that \(\phi_k\circ \text{Aut}_{\mathrm{CR}}(M, J_k)\circ \phi_k^{-1}\) is a Lie subgroup of \(\text{Aut}_{\mathrm{CR}}(M, J_0)\).
scientific article; zbMATH DE number 6632727