Title: CR automorphisms of real analytic manifolds in complex space
Abstract: In this paper we shall give sufficient conditions for local CR diffeomorphisms between two real analytic submanifolds of C^ to be determined by finitely many derivatives at finitely many points.These conditions will also be shown to be necessary in model cases.We shall also show that under the same conditions, the Lie algebra of the infinitesimal CR automorphisms at a point is finite dimensional.Let M be a real analytic submanifold of C^.For p € M a CR vector at p is a vector of the form X^i c j'^~^ c j ^ ^o tangent to M at p.If M 1 J oZj is another submanifold of C^, a mapping F : M -> M' is called CR if for any p 6 M the pushforward F*X of any CR vector X on M at p is a CR vector of M 1 at F{p).In particular, the restriction to M of a germ of a holomorphic diffeomorphism H from C^ to itself is a CR map from M to its image.As in [BER1] (see Stanton [Stl] for the case of a hypersurface), we shall say that a real submanifold of C^ is holomorphically nondegenerate if there is no germ of a nontrivial vector field X)j=i c j(^)w^^ w ^h c j(^) holomorphic, tangent to M. If M is holomorphically nondegenerate there is an integer Z(M), with 0 < l(M) < iV, called the Levi number of M (see §1) which measures the holomorphic nondegeneracy of M. If M is a Levi-nondegenerate hypersurface then l(M) = 1.A connected real analytic submanifold is minimal almost everywhere if there is no germ of a holomorphic function whose restriction to M is a nonconstant real-valued function.This coincides with the notion of being minimal at most points in the sense of Tumanov [Tul].