Volume-Minimizing Foliations on Spheres |
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Authors: | Fabiano Brito David L Johnson |
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Institution: | (1) Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de Sāo Paulo, R. do Matāo 1010, 05508-900 Sāo Paulo-SP, Brazil;(2) Department of Mathematics, Lehigh University, 14 E. Packer Avenue, Bethlehem, PA, 18015, U.S.A |
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Abstract: | The volume of a k-dimensional foliation
in a Riemannian manifold Mn is defined as the mass of the image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing the construction by Gluck and Ziller (Comment. Math. Helv. 61 (1986), 177–192), ‘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, even though they are not homologous to the graph of a foliation, provide lower bounds for volumes of regular three-dimensional foliations of S4n+3 and regular seven-dimensional foliations of S15, since the double of these currents will be homologous to twice the graph of any smooth foliation by 3-manifolds.The second author was supported during this research by grants from the Universidade de Sāo Paulo, FAPESP Proc. 1999/02684-5, and Lehigh University, and thanks those institutions for enabling the collaboration involved in this work.Mathematics Subject Classifications (2000). 53C12, 53C38. |
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Keywords: | foliations spheres calibrations |
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