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Area-minimizing vector fields on round 2-spheres. (English) Zbl 1187.53029

Summary: 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 3 [H. Gluck and W. Ziller, Comment. Math. Helv. 61, 177–192 (1986; Zbl 0605.53022)]. 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 \(\mathbb RP^{2}\) 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.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Citations:

Zbl 0605.53022
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References:

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