On the volume of a unit vector field in 3 dimensions via calibrations

Abstract

We give a new proof of the well-known result that the minimal volume vector fields on S^3(r) are the Hopf vector fields. Such proof relies on calibration theory, arising here from a systematic view given by a natural source of differential forms. Our results serve in particular for all r. A classification of relevant calibrations on T^1M for every oriented 3-manifold M of constant sectional curvature is given, continuing the study of the usual fundamental differential system of Riemannian geometry. Showing applications of this striking differential system is one of the purposes of this article. We deduce new properties of the geodesic flow vector field of space forms, which interacts with the solutions of the minimal volume problem both in elliptic and hyperbolic geometry, in any dimension. The solution – unknown – for the hyperbolic case in 3-dimensions being most dependent on the homology class of the domain and boundary values of the vector fields. This is illustrated with a noteworthy example which ironically works just for curvature -1.