Abstract
For \(n\ge 1\), we exhibit a lower bound for the volume of a unit vector field on \({\mathbb {S}}^{2n+1}\backslash \{\pm p\}\) depending on the absolute values of its Poincaré indices around \(\pm p\). We determine which vector fields achieve this volume, and discuss the idea of having multiple isolated singularities of arbitrary configurations.
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References
Borrelli, V., Gil-Medrano, O.: Area-minimizing vector fields on round \(2\)-spheres. J. Reine Angew. Math. 640, 85–99 (2010)
Borrelli, V., Gil-Medrano, O.: A critical radius for unit Hopf vector fields on spheres. Math. Ann. 334(4), 731–751 (2006)
Brito, F.G.B., Chacón, P.M.: A topological minorization for the volume of vector fields on 5-manifolds. Arch. Math. 85, 283–292 (2005)
Brito, F.G.B., Chacón, P.M., Johnson, D.L.: Unit vector fields on antipodally punctured spheres: big index, big volume. Bull. Soc. Math. Fr. 136(1), 147–157 (2008)
Brito, F.B., Chacón, P.M., Naveira, A.M.: On the volume of unit vector fields on spaces of constant sectional curvature. Comment Math. Helv. 79, 300–316 (2004)
Chacón, P.M., Nunes, G.S.: Energy and topology of singular unit vector fields on \({\mathbb{S}}^3\). Pac. J. Math. 231(1), 27–34 (2007)
Chern, S.S.: A simple intrinsic proof of the Gauss–Bonnet formula for closed Riemannian manifolds. Ann. Math. 45(4), 747–752 (1944)
Chern, S.S.: On the curvatura integra in a Riemannian manifold. Ann. Math. 46(2), 674–684 (1945)
Gluck, H., Ziller, W.: On the volume of a unit field on the three-sphere. Comment Math. Helv. 61, 177–192 (1986)
Johnson, D.L.: Volume of flows. Proc. Am. Math. Soc. 104, 923–932 (1988)
Pedersen, S.L.: Volume of vector fields on spheres. Trans. Am. Math. Soc. 336, 69–78 (1993)
Reznikov, A.G.: Lower bounds on volumes of vector fields. Arch. Math. 58, 509–513 (1992)
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The authors would like to thank the anonymous reviewers for their helpful and constructive comments that greatly contributed to improving the final version of the paper.
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Icaro Gonçalves: Supported by a scholarship from the National Postdoctoral Program, PNPD-CAPES. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.
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Brito, F.G.B., Gomes, A.O. & Gonçalves, I. Poincaré index and the volume functional of unit vector fields on punctured spheres. manuscripta math. 161, 487–499 (2020). https://doi.org/10.1007/s00229-019-01107-y
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DOI: https://doi.org/10.1007/s00229-019-01107-y