Parametrization of local CR automorphisms by finite jets and applications

Abstract

For any real-analytic hypersurface M in complex euclidean space of dimension >= 2 which does not contain any complex-analytic subvariety of positive dimension, we show that for every point p in M the local real-analytic CR automorphisms of M fixing p can be parametrized real-analytically by their l(p)-jets at p. As a direct application, we derive a Lie group structure for the topological group Aut(M,p). Furthermore, we also show that the order l(p) of the jet space in which the group Aut(M,p) embeds can be chosen to depend upper-semicontinuously on p. As a first consequence, it follows that that given any compact real-analytic hypersurface M in complex euclidean space, there exists an integer k depending only on M such that for every point p in M germs at p of CR diffeomorphisms mapping M into another real-analytic hypersurface in a complex space of the same dimension are uniquely determined by their k-jet at that point. Another consequence is a boundary version of H. Cartan's uniqueness theorem. Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.