Foliations by minimal surfaces and contact structures on certain closed 3‐manifolds

Abstract

Let (M, g) be a closed, connected, oriented C∞ Riemannian 3‐manifold with tangentially oriented flow F. Suppose that F admits a basic transverse volume form μ and mean curvature one‐form κ which is horizontally closed. Let {X, Y} be any pair of basic vector fields, so μ(X, Y) = 1. Suppose further that the globally defined vector 𝒱[X, Y] tangent to the flow satisfies [Z.𝒱[X, Y]] = fZ𝒱[X, Y] for any basic vector field Z and for some function fZ depending on Z. Then, 𝒱[X, Y] is either always zero and H, the distribution orthogonal to the flow in T(M), is integrable with minimal leaves, or 𝒱[X, Y] never vanishes and H is a contact structure. If additionally, M has a finite‐fundamental group, then 𝒱[X, Y] never vanishes on M, by the above together with a theorem of Sullivan (1979). In this case H is always a contact structure. We conclude with some simple examples.