Volume-Minimizing Foliations on Spheres

Abstract

The volume of a k-dimensional foliation ${\cal F}$ in a Riemannian manifold Mⁿ is defined as the mass of the image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing the construction by Gluck and Ziller (Comment. Math. Helv. 61 (1986), 177–192), ‘singular’ foliations by 3-spheres are constructed on round spheres S⁴ⁿ⁺³, as well as a singular foliation by 7-spheres on S¹⁵, which minimize volume within their respective relative homology classes. These singular examples, even though they are not homologous to the graph of a foliation, provide lower bounds for volumes of regular three-dimensional foliations of S⁴ⁿ⁺³ and regular seven-dimensional foliations of S¹⁵, since the double of these currents will be homologous to twice the graph of any smooth foliation by 3-manifolds.