Unit vector fields on antipodally punctured spheres: big index, big volume

Fabiano G. B. Brito; Pablo M. Chacón; David L. Johnson

Bulletin de la Société Mathématique de France (2008)

  • Volume: 136, Issue: 1, page 147-157
  • ISSN: 0037-9484

Abstract

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We establish in this paper a lower bound for the volume of a unit vector field v → defined on 𝐒 n ∖ { ± x } , n = 2 , 3 . This lower bound is related to the sum of the absolute values of the indices of v → at x and - x .

How to cite

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Brito, Fabiano G. B., Chacón, Pablo M., and Johnson, David L.. "Unit vector fields on antipodally punctured spheres: big index, big volume." Bulletin de la Société Mathématique de France 136.1 (2008): 147-157. <http://eudml.org/doc/272456>.

@article{Brito2008,
abstract = {We establish in this paper a lower bound for the volume of a unit vector field $\vec\{v\}$ defined on $\textbf \{S\}^n\setminus \lbrace \pm x\rbrace $, $n=2,3$. This lower bound is related to the sum of the absolute values of the indices of $\vec\{v\}$ at $x$ and $-x$.},
author = {Brito, Fabiano G. B., Chacón, Pablo M., Johnson, David L.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {unit vector fields; volume; singularities; index},
language = {eng},
number = {1},
pages = {147-157},
publisher = {Société mathématique de France},
title = {Unit vector fields on antipodally punctured spheres: big index, big volume},
url = {http://eudml.org/doc/272456},
volume = {136},
year = {2008},
}

TY - JOUR
AU - Brito, Fabiano G. B.
AU - Chacón, Pablo M.
AU - Johnson, David L.
TI - Unit vector fields on antipodally punctured spheres: big index, big volume
JO - Bulletin de la Société Mathématique de France
PY - 2008
PB - Société mathématique de France
VL - 136
IS - 1
SP - 147
EP - 157
AB - We establish in this paper a lower bound for the volume of a unit vector field $\vec{v}$ defined on $\textbf {S}^n\setminus \lbrace \pm x\rbrace $, $n=2,3$. This lower bound is related to the sum of the absolute values of the indices of $\vec{v}$ at $x$ and $-x$.
LA - eng
KW - unit vector fields; volume; singularities; index
UR - http://eudml.org/doc/272456
ER -

References

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  1. [1] V. Borrelli & O. Gil-Medrano – « Area minimizing vector fields on round 2 -spheres », 2006, preprint. Zbl1187.53029MR2629689
  2. [2] —, « A critical radius for unit Hopf vector fields on spheres », Math. Ann.334 (2006), p. 731–751. Zbl1115.53025MR2209254
  3. [3] F. G. B. Brito – « Total bending of flows with mean curvature correction », Differential Geom. Appl.12 (2000), p. 157–163. Zbl0995.53023MR1758847
  4. [4] F. G. B. Brito & P. M. Chacón – « A topological minorization for the volume of vector fields on 5-manifolds », Arch. Math. (Basel) 85 (2005), p. 283–292. Zbl1089.53025MR2172387
  5. [5] F. G. B. Brito, P. M. Chacón & A. M. Naveira – « On the volume of unit vector fields on spaces of constant sectional curvature », Comment. Math. Helv.79 (2004), p. 300–316. Zbl1057.53022MR2059434
  6. [6] P. M. Chacón – « Sobre a energia e energia corrigida de campos unitários e distribuições. Volume de campos unitários », Thèse, Universidade de São Paulo, Brazil, 2000, and Universidad de Valencia, Spain, 2001. 
  7. [7] S. S. Chern – « On the curvatura integra in a Riemannian manifold », Ann. of Math. (2) 46 (1945), p. 674–684. Zbl0060.38104MR14760
  8. [8] S. S. Chern & J. Simons – « Characteristic forms and geometric invariants », Ann. of Math. (2) 99 (1974), p. 48–69. Zbl0283.53036MR353327
  9. [9] O. Gil-Medrano & E. Llinares-Fuster – « Second variation of volume and energy of vector fields. Stability of Hopf vector fields », Math. Ann.320 (2001), p. 531–545. Zbl0989.53020MR1846776
  10. [10] H. Gluck & W. Ziller – « On the volume of a unit vector field on the three-sphere », Comment. Math. Helv.61 (1986), p. 177–192. Zbl0605.53022MR856085
  11. [11] D. L. Johnson – « Chern-Simons forms on associated bundles, and boundary terms », Geometria Dedicata120 (2007), p. 23–24. Zbl1148.53014MR2350146
  12. [12] S. L. Pedersen – « Volumes of vector fields on spheres », Trans. Amer. Math. Soc.336 (1993), p. 69–78. Zbl0771.53023MR1079056

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