Global solvability and global hypoellipticity of complex vector fields on surfaces

Abstract

We consider smooth, nonvanishing complex (not essentially real) vector fields L=X+iY that satisfy the Nirenberg-Treves condition (P) and are allowed to possess some closed one-dimensional orbits in the sense of Sussmann that we call clean orbits. Concerning global solvability, we deal with two different cases —depending on whether the surface is compact or not— and prove two positive results which extend and unify several known results on the subject.In our positive results, it is assumed that X∧Y vanishes of order greater than 1 on all closed one-orbits, on the other hand, we prove that global solvability cannot hold if there exists at least one closed one-orbit on which X∧Y vanishes of order 1.We also give a characterization of globally hypoelliptic complex vector fields L in terms of the topological type of the equivalence classes determined by a standard equivalence relation defined on the set of non elliptic points of L. Some of the consequences are that globally hypoelliptic vector fields are globally solvable and that if a surface M carries a globally hypoelliptic vector field, it must be parallelizable.