Study of overdetermined system of complex vector fields

Abstract

We have investigated the local qualitive behavior of a certain formally integrable system of complex vector fields defined on an open subset of the N-dimensional Euclidean space. First, we deal with the local integrabillity of the so called CR-hypersurfaces. After reducing the corresponding CR-structure into a canonical form in some sense via a local change of coordinates, we have found a necessary condition and a sufficient condition for a given CR-hypersurface to be locally integrable. These results explain the nonlocal integrability of now calssical example of L. Nirenberg and extend the result of N. Hanges for N=3. It also supplement the Kuranishi``s theorem saying that every CR-hypersurface is locally integrable if N is greater than or equals to 9. Secondly, we deal with the hypoellipticity and analytic-hypoellpticity of a locally integrable system of n=N-1 complex vector fields. Using the local constnacy principle of such a system due to F. Treves and M. Baouendi, we have found sufficient conditions for such a system to be (anslytic-) hypoelliptic. In some restricted cases, these conditions are shown to be necessary and sufficient. Fundamental idea of the proof of these results is to form the Laplacian associated to the system, which is a second order differenteal operator with double characteristicts and to observe that hypoellipticity of the Laplacian implies that of the system. Finally, we provide a brief survey of the solvabliity of the system, the one of the fundamental question closely related to the hypoellipticity and some concrete examples explaining the scope of this work.