On foliations defined by harmonic functions

Abstract

Riemann surfaces. A two-dimensional connected manifold S is called a Riemann surface if S is equipped with a complex atlas {Uα, zα} with biholomorphic functions zαz β (on their domains). The maximal atlas defines a conformal structure on S. Riemann surfaces S1 and S2 are said to be conformally equivalent if there is a conformal homeomorphism φ : S1 → S2 with conformal inverse. The complex plane C is a Riemann surface. Its conformal structure is determined by the atlas whose only chart is the identity mapping C → C. Every open connected domain D ⊆ C also becomes a Riemann surface in a natural way. The famous uniformization theorem claims that every noncompact simply connected Riemann surface is conformally equivalent to either C or the upper half-plane, which is conformally equivalent to the open unit diskD1 = {|z| < 1}. Let S be a Riemann surface with conformal structure {Uα, zα}. A function f : S → R is said to be harmonic on S if, for any chart Uα, the function f ◦ z−1 α is harmonic on zα(Uα). Kaplan [1] obtained the following remarkable result. Let F be a foliation without singularities on the plane R2. In this case, there are a conformal structure transforming R2 to a Riemann surface S and a harmonic function f on S such that the leaves of the foliation F are level curves of the function f . Following [1], we say that a foliation F is parabolic (hyperbolic, respectively) if S is conformally equivalent to C (to D1, respectively). Kaplan [1] proved that every foliation without singularities on the