A Cauchy-Kowalevsky theorem for overdetermined systems of nonlinear partial differential equations and geometric applications.

Abstract

The main motivation for the work presented in this paper is to construct real hypersurfaces in Cn+1 with maximal Levi number (see below for the definition), a problem that has been open since Levi numbers were introduced in [BHR96]. The examples constructed here are tube hypersurfaces. Moreover, we give a local description of all such hypersurfaces. In order for the real hypersurface M to have the desired properties, Σ must be a non-cylindrical hypersurface whose Gauss map (or equivalently second fundamental form) has rank one. To construct locally defined hypersurfaces in Rn+1 with these properties, indeed to parametrize all such, we prove an existence and uniqueness theorem concerning a Cauchy problem for a class of overdetermined systems of nonlinear partial di erential equations in Rn.