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

TL;DR

This paper establishes a Cauchy-Kowalevsky theorem for overdetermined nonlinear PDE systems, providing solvability conditions and applying them to classify certain geometric structures in complex space.

Contribution

It extends the classical Cauchy-Kowalevsky theorem to overdetermined systems and applies the results to classify real tube hypersurfaces with maximal Levi number.

Findings

Derived necessary and sufficient conditions for solvability of overdetermined PDE systems.

Classified all real tube hypersurfaces in complex space with maximal Levi number.

Abstract

We consider a Cauchy problem for an overdetermined system of PDEs, and give necessary and sufficient conditions for solvability of this Cauchy problem for all data. As an application, we find all real tube hypersurfaces in complex space whose Levi number is maximal.