Volume-minimizing foliations on spheres
Brito, Fabiano · Johnson, David L.
Original · EN
The volume of a k-dimensional foliation F in a Riemannian manifold Mⁿ is defined as the mass of image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing a construction by Gluck and Ziller, "singular" foliations by 3-spheres are constructed on round spheres S⁴ⁿ⁺³, as well as a singular foliation by 7-spheres on S¹⁵, which minimize volume within their respective relative homology classes. These singular examples provide lower bounds for volumes of regular 3-dimensional foliations of S⁴ⁿ⁺³ and regular 7-dimensional foliations of S¹⁵.
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