Holomorphic support functions for uniformly pseudoconvex hypersurfaces, with an application to CR maps

Abstract

We construct holomorphic support functions for smooth weakly pseudoconvex hypersurfaces with Levi form of constant rank. These are then applied to show that formal holomorphic curves which are tangential to infinite order to such a hypersurface must be formally contained in its Levi foliation. As a consequence, we obtain a holomorphic deformation theorem for nowhere smooth CR maps into smooth pseudoconvex hypersurfaces with one-dimensional Levi foliation, strengthening a very general result of Lamel and Mir about formal deformations in this particular case.