Area-minimizing vector fields on round 2-spheres
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Vincent Borrelli
Abstract
A vector field V on an n-dimensional round sphere Sn(r) defines a submanifold V(Sn) of the tangent bundle TSn. The Gluck and Ziller question is to find the infimum of the n-dimensional volume of V(Sn) 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 S2, 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 T1S2(r). We prove that if this minimal surface is homeomorphic to ℝP2 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.
© Walter de Gruyter Berlin · New York 2010
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- On the torsion of Drinfeld modules of rank two
- Endpoint maximal and smoothing estimates for Schrödinger equations
- Kähler-Ricci flow on stable Fano manifolds
- Area-minimizing vector fields on round 2-spheres
- Koppelman formulas on Grassmannians
- A quotient restriction theorem for actions of real reductive groups
- Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence
- On the bounded cohomology of semi-simple groups, S-arithmetic groups and products
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Articles in the same Issue
- On the torsion of Drinfeld modules of rank two
- Endpoint maximal and smoothing estimates for Schrödinger equations
- Kähler-Ricci flow on stable Fano manifolds
- Area-minimizing vector fields on round 2-spheres
- Koppelman formulas on Grassmannians
- A quotient restriction theorem for actions of real reductive groups
- Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence
- On the bounded cohomology of semi-simple groups, S-arithmetic groups and products
- Über Pro-p-Fundamentalgruppen markierter arithmetischer Kurven
