Minimality and perturbations of CR manifolds.

Abstract

A smooth, generic CR submanifold M of C$\sp{n}$ is said to be minimal at a point p if there is no germ of a proper submanifold of M, containing p, of the same CR dimension as M. Minimality is equivalent to uniform, holomorphic wedge-extendibility of locally-defined CR functions on M, and can be characterized in terms of the notion of defect introduced by A. Tumanov. We study the stability of the minimality condition under small perturbations of the manifold M, by studying the effect of the perturbations on the defect at the point p. A density result is obtained, that manifolds minimal at p form a dense subset of the set of all manifolds through p. As an application, we give elementary geometric proofs of the most general results regarding propagation of wedge-extendibility of CR functions along certain subsets of M, including those of N. Hanges and J. Sjostrand, and J.-M. Trepreau.