Volume-Minimizing Foliations on Spheres

Abstract

The volume of a k-dimensional foliation F in a Riemannian manifold M n 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 S 4nþ3 , as well as a singular foliation by 7-spheres on S 15 , 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 S 4nþ3 and regular seven-dimensional foliations of S 15 , since the double of these currents will be homologous to twice the graph of any smooth foliation by 3-manifolds. (2000). 53C12, 53C38. Mathematics Subject ClassificationsIn [4], Herman Gluck and Wolfgang Ziller asked which foliations were 'best-organized', in that an energy functional they called the volume was minimized. The volume of a foliation is the mass of the image of the Gauss map, which in the case of a one-dimensional foliation is the mass of the unit tangent flow field in T 1 ðMÞ.They were able to show that the standard one-dimensional foliation (or flow, in their terminology) of S 3 by the fibers of the Hopf fibration S 3 ! S 2 minimized volume among all foliations of the round S 3 . Their method of proof, involving calibrations, did not generalize, however.It is not the case that even the most obvious generalization of Gluck and Ziller's example to higher dimensions, the Hopf fibration S 5 ! CP 2 , is volume-minimizing [5]. Sharon Pedersen showed in her thesis that there was a foliation of S 5 with much less volume than the Hopf fibration, although her example is singular [8]. It may well be that the volume-minimizing one-dimensional foliations on S 5 is singular, although ?