International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 21, Pages 1323-1330

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

Richard H. Escobales Jr.

Department of Mathematics and Statistics, Canisius College, Buffalo 14208, NY, USA

Received 14 July 2002; Revised 7 August 2002

Copyright © 2003 Richard H. Escobales. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.