Transversality and local geometric properties of CR mappings

Peter Ebenfelt Department of Mathematics, University of California, San Diego

ABSTRACT: A classical result (due to Poincare,...) states that any nonconstant holomorphic mapping sending a piece of the unit sphere in $\mathbb C^n$ into another piece of the unit sphere extends as a global automorphism of the sphere (in fact of the whole ball). In particular, the mapping is necessarily a local diffeomorphism at every point. It turns out that part of this phenomenon (i.e. the maximal rank of the transversal part of the differential) carries over to a much more general setting in which the spheres are replaced by CR manifolds of finite (commutator) type. Having maximal rank for the tangential part of the differential is a more subtle question. In this talk, we shall discuss some recent results and open questions along these lines.