Total curvature and volume of foliations on the sphere S 2

Amine Fawaz

Open Mathematics (2009)

  • Volume: 7, Issue: 4, page 660-669
  • ISSN: 2391-5455

Abstract

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In this paper we study a curvature integral associated with a pair of orthogonal foliations on the Riemann sphere S 2 and we compute the minimal value of the volume of meromorphic foliations.

How to cite

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Amine Fawaz. "Total curvature and volume of foliations on the sphere S 2." Open Mathematics 7.4 (2009): 660-669. <http://eudml.org/doc/269540>.

@article{AmineFawaz2009,
abstract = {In this paper we study a curvature integral associated with a pair of orthogonal foliations on the Riemann sphere S 2 and we compute the minimal value of the volume of meromorphic foliations.},
author = {Amine Fawaz},
journal = {Open Mathematics},
keywords = {Foliation; Geodesic curvature; Green’s function; Singular; Volume; foliation; geodesic curvature; Green's function; singular; volume},
language = {eng},
number = {4},
pages = {660-669},
title = {Total curvature and volume of foliations on the sphere S 2},
url = {http://eudml.org/doc/269540},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Amine Fawaz
TI - Total curvature and volume of foliations on the sphere S 2
JO - Open Mathematics
PY - 2009
VL - 7
IS - 4
SP - 660
EP - 669
AB - In this paper we study a curvature integral associated with a pair of orthogonal foliations on the Riemann sphere S 2 and we compute the minimal value of the volume of meromorphic foliations.
LA - eng
KW - Foliation; Geodesic curvature; Green’s function; Singular; Volume; foliation; geodesic curvature; Green's function; singular; volume
UR - http://eudml.org/doc/269540
ER -

References

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  1. [1] Aubin T., Some nonlinear problems in Riemannian geometry, Springer-Verlag, Berlin, 1998 
  2. [2] Fabiano B., Pablo M.C., Naveira A.M., On the volume of unit vector fields on spaces of constant curvature, Comment. Math. Helv., 2004, 79, 300–316 http://dx.doi.org/10.1007/s00014-004-0802-4[Crossref] Zbl1057.53022
  3. [3] Fawaz A., Energy and foliations on Riemann surfaces, Ann. Global Anal. Geom., 2005, 28, 75–89 http://dx.doi.org/10.1007/s10455-005-4405-0[Crossref] 
  4. [4] Fathi A., Laudenbach F., Poenaru V., Travaux de Thurston sur les surfaces, Astérisque, 1979, vol. 66–67, SMF Paris (in French) 
  5. [5] Gutkin E., Newton P., The method of images and Green’s function for spherical domains, J. Phys. A: Math. Gen., 2004, 37, 11989–12003 http://dx.doi.org/10.1088/0305-4470/37/50/004[Crossref] Zbl1067.78005
  6. [6] Gil-Medrano O., Llinares-Fuster E., Second variation of volume and energy of vector fields. Stability of Hopf vector fields, Math. Ann., 2001, 320, 531–545 http://dx.doi.org/10.1007/PL00004485[Crossref] Zbl0989.53020
  7. [7] Gil-Medrano O., Llinares-Fuster E., Minimal unit vector fields, Tohoku Math. J., 2002, 54, 71–84 http://dx.doi.org/10.2748/tmj/1113247180[Crossref] Zbl1006.53053
  8. [8] Gluck H., Ziller W., On the volume of a unit vector field on the three sphere, Comment. Math. Helv., 1986, 61, 177–192 http://dx.doi.org/10.1007/BF02621910[Crossref] Zbl0605.53022
  9. [9] Johnson L.D., Volumes of flows, Proc. Amer. Math. Soc., 1988, 104, 923–932 http://dx.doi.org/10.2307/2046818[Crossref] Zbl0687.58031
  10. [10] Langevin R., Levitt G., Courbure totale des feuilletages des surfaces, Comment. Math. Helv., 1982, 57, 175–195 (in French) http://dx.doi.org/10.1007/BF02565855[Crossref] Zbl0502.53026
  11. [11] Pavlov V., Buisine D., Goncharov V., Formation of vortex clusters on a sphere, Nonlinear processes in geophysics, 2001, 8, 9–19 
  12. [12] Pedersen L.S., Volumes of vector fields on spheres, Trans. Amer. Math. Soc., 1993, 336, 69–78 http://dx.doi.org/10.2307/2154338[Crossref] 
  13. [13] Thurston W., Three-dimensional geometry and topology, 1, Princeton University Press, 1997 Zbl0873.57001
  14. [14] Tondeur P., Geometry of foliations, Monogr. Math., 90, Birkhäuser, 1997 Zbl0905.53002

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