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Let S be a generic C-infinity smooth CR manifold in C^n, n > 1, and let M bea generic C-infinity CR submanifold of S X C^m. We prescribe conditions on M sothat it is the disjoint union of graphs of CR maps f:S-->C^m. We also considerthe special case where S is the boundary of a bounded, smoothly bounded openset D in C^n, n > 1. In this case we obtain conditions which guarantee thatsuch an f above extends to be analytic on D. This provides a solution to aparticular nonlinear Riemann-Hilbert boundary value problem for analyticfunctions.

Foliation by graphs of CR mappings and a nonlinear Riemann-Hilbert problem for smoothly bounded domains