Foliation by Graphs of Cr Mappings and a Nonlinear Riemann-hilbert Problem for Smoothly Bounded Domains

Let S be a generic C∞ CR manifold in C, l ≥ 2, and let M be a generic C∞ CR submanifold of S × C, m ≥ 1. We prescribe conditions on M so that it is the disjoint union of graphs of CR maps f : S → C. We also consider the special case where S is the boundary of a bounded, smoothly bounded open set D in C, l ≥ 2. In this case we obtain conditions which guarantee that such an f above extends to be analytic on D. This provides a solution to a particular nonlinear Riemann-Hilbert boundary value problem for analytic functions. Let D be a bounded, smoothly bounded domain in C, l ≥ 2, and let M be a real C submanifold of ∂D×C, m ≥ 1. We here address the question of when there exists a mapping f such that (RH) [ f : D → C is continuous on D and analytic on D such that the graph of f over ∂D is contained in M . A problem where one is required to find an f satisfying (RH) is often called a RiemannHilbert problem; Riemann proposed such a question for l = m = 1 in 1851. We shall refer to the problem of finding an f satisfying (RH) as the Riemann-Hilbert problem for M . If an f exists satisfying (RH) then we shall say that the Riemann-Hilbert problem for M is solvable and that f is a solution. For z ∈ ∂D, let Mz ≡ {w ∈ C : (z, w) ∈ M}. We will say here that the Riemann-Hilbert problem (RH) is linear if for every z ∈ ∂D, Mz is a real affine subspace of C. We shall say that the Riemann-Hilbert problem (RH) is nonlinear if it is not linear. For the case l = 1 we mention a few references:[V,Sh1,Sh2,Fo,S,HMa,We1-5,B1,B2]. See [We4] for a useful survey and reference list. For l ≥ 2, see [B1,B3,BD,D1,D2]. We will first address the following more general question. Let S be a C CR manifold in C (e.g., a real hypersurface in C) and let M be a real C submanifold of S × C, m ≥ 1. Let z ∈ S and U a neighborhood of z in S. Does there exist a CR map f : U → C whose 1991 Mathematics Subject Classification. 32A40, 32V10, 32H12.