A criterion for the nonexistence of closed leaves in unorientable planar foliations

The family of all smooth foliations F on an open set Ω ⊂ R2 ∼= C is naturally parameterized by all smooth maps X : Ω → S1 = {z ∈ C : |z| = 1}, in the sense that the values ± √ X(p) determine the tangent line to the leaf of F at p ∈ Ω. If F is further assumed to be orientable, a smooth global branch Y of the square root of X can be chosen. In this case, one has the classical Lyapunov criterion: if there is a real-valued u ∈ C1(Ω) such that Y u = Re { 2Y uz̄ } is nowhere zero, then F has no closed leaves (the vector field Y has no periodic orbits). In this paper we introduce an analytic criterion for the nonexistence of closed leaves, similar in spirit to that of Lyapunov, but which allows for F to be unorientable as well. The possible lack of orientability makes the replacement for the first order differential operator Y considerably more involved. In fact, one has to work with a second order linear hyperbolic differential operator LF whose coefficients carry information about the curvature of the leaves of F . It is shown that if F is given by X : Ω → S1 in the manner described above, and there exists a real-valued u ∈ C2(Ω) such that LFu := Im{2Xuz̄z̄+ ( Xz −XXz ) uz̄} is nowhere zero, then F has no closed leaves. We apply the new criterion when X is holomorphic, providing also an example that shows the need for the first order term in the definition of LF .