Volumes of Vector Fields on Spheres

Abstract

In this paper we study the problem: What is the unit vector field of smallest volume on an odd-dimensional sphere? We exhibit on each sphere a unit vector field with singularity which has exceptionally small volume on spheres of dimension greater than four. We conjecture that this volume is the infimum for volumes of bona fide unit vector fields, and is only achieved by the singular vector field. We generalize the construction of the singular vector field to give a family of cycles in Stiefel manifolds, each of which is a smooth manifold except for one singular point. Except for some low-dimensional cases, the tangent cones at these singular points are volume-minimizing; and half of the cones are nonorientable. Thus, we obtain a new family of nonorientable volume-minimizing cones.