Area-minimizing vector fields on round 2-spheres

Abstract

A vector field V on an n -dimensional round sphere S n ( r ) defines a submanifold V ( S n ) of the tangent bundle TS n . The Gluck and Ziller question is to find the infimum of the n -dimensional volume of V ( S n ) among unit vector fields. This volume is computed with respect to the natural metric on the tangent bundle as defined by Sasaki. Surprisingly, the problem is only solved for dimension three [Gluck and Ziller, Math. Helv. 61: 177–192, 1986]. In this article we tackle the question for the 2-sphere. Since there is no globally defined vector field on S 2 , the infimum is taken on singular unit vector fields without boundary. These are vector fields defined on a dense open set and such that the closure of their image is a surface without boundary. In particular if the vector field is area-minimizing it defines a minimal surface of T 1 S 2 ( r ). We prove that if this minimal surface is homeomorphic to ℝ P 2 then it must be the Pontryagin cycle . It is the closure of unit vector fields with one singularity obtained by parallel translating a given vector along any great circle passing through a given point. We show that Pontryagin fields of the unit 2-sphere are area-minimizing.