Levi number tube hypersurfaces Cauchy Hypersurface Cauchy-Kowalevsky differential equation
Taylor & Francis
Baouendi, M. S.; Ebenfelt, P.; Zaitsev, D., A Cauchy-Kowalevsky theorem for overdetermined systems of nonlinear partial differential equations and geometric applications., Communications in Partial Differential Equations, 34, 10, 2009, 1180--1207
Communications in Partial Differential Equations;34, 10
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.
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