Boundary problem for Levi flat graphs

Original · EN

In an earlier paper the authors provided general conditions on a real codimension 2 submanifold S⊂ Cⁿ, n≥ 3, such that there exists a possibly singular Levi-flat hypersurface M bounded by S. In this paper we consider the case when S is a graph of a smooth function over the boundary of a bounded strongly convex domain Ω⊂ Cⁿ⁻¹× R and show that in this case M is necessarily a graph of a smooth function over Ω. In particular, M is non-singular.