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

TL;DR

This paper provides a new calibration-based proof that Hopf vector fields minimize volume on 3-spheres, extends calibration classification to all space forms, and explores applications to geodesic flows in constant curvature 3-manifolds.

Contribution

It introduces a systematic calibration approach to minimal volume vector fields on spheres and space forms, and analyzes geodesic flow properties in these geometries.

Findings

Hopf vector fields are volume minimizers on $oldsymbol{S^3(r)}$.

Classification of calibrations on $T^1M$ for all constant curvature 3-manifolds.

New properties of geodesic flow vector fields in space forms.

Abstract

We give a new proof of the well-known result that the minimal volume vector fields on $\mathbb{S}^3(r)$ are the Hopf vector fields. Such proof relies again on calibration theory, arising here from a systematic point of 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 \textit{usual} fundamental differential system of Riemannian geometry. Showing applications of this differential system is also 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 $-1$.