Poincaré index and the volume functional of unit vector fields on punctured spheres

  • Fabiano G. B. Brito
  • André O. Gomes
  • Icaro GonçalvesEmail author


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.

Mathematics Subject Classification

53C20 57R25 53C12 


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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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Centro de Matemática, Computação e CogniçãoUniversidade Federal do ABCSanto AndréBrazil
  2. 2.Dpto. de Matemática, Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil

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