Volumes of vector fields on spheres. (English) Zbl 0771.53023

This paper contains results from the author’s Ph.D. thesis and centers around the problem of determining the unit vector fields on an odd- dimensional round unit sphere which are optimal, that is, the submanifold of the unit sphere bundle determined by this field has minimal volume. The starting point is the nice result of H. Gluck and W. Ziller who proved that the Hopf vector field associated to the Hopf fibration of \(S^ 3\) is the unique optimal one and the further result of D. Johnson who proved that for \(S^ 5\) the Hopf vector fields are only unstable critical points of the volume. The author first shows that on \(S^{2k+1}\), \(k \geq 2\), there exist unit vector fields of exceptionally small volume which converge to a vector field with one singularity. This result suggests the author to conjecture that there are no optimal unit vector fields on \(S^{2k+1}\), \(k\geq 2\), and that the limiting field with singularity is of minimum volume in its homology class in the unit tangent bundle. The construction of the singular vector field is also generalized to give a singular field, with one singular point, of orthonormal \((k-1)\)-frames on \(S^ n\). Moreover, it is shown that, except for some low dimensions, the tangent cones at these singular points are volume-minimizing. In this way the author provides a new family of non-orientable volume-minimizing cones.


53C20 Global Riemannian geometry, including pinching
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
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