Foliations by minimal surfaces and contact structures on certain closed 3-manifolds
Summary: Let \((M,g)\) be a closed, connected, oriented \(C^\infty\) Riemannian 3-manifold with tangentially oriented flow \(\mathbf{F}\). Suppose that \(\mathbf{F}\) admits a basic transverse volume form \(\mu\) and mean curvature one-form \(\kappa\) which is horizontally closed. Let \(\{X,Y\}\) be any pair of basic vector fields, so \(\mu(X,Y)=1\). Suppose further that the globally defined vector \(\mathcal{V}[X,Y]\) tangent to the flow satisfies \([Z,\mathcal{V}[X,Y]]=f_{Z}\mathcal{V}[X,Y]\) for any basic vector field \(Z\) and for some function \(f_Z\) depending on \(Z\). Then, \(\mathcal{V}[X,Y]\) is either always zero and \(\mathbf{H}\), the distribution orthogonal to the flow in \(T(M)\), is integrable with minimal leaves, or \(\mathcal{V}[X,Y]\) never vanishes and \(\mathbf{H}\) is a contact structure. If additionally, \(M\) has a finite-fundamental group, then \(\mathcal{V}[X,Y]\) never vanishes on \(M\), by the above together with a theorem of Sullivan (1979). In this case \(\mathbf{H}\) is always a contact structure. We conclude with some simple examples.
scientific article; zbMATH DE number 1930031