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.
@article{bwmeta1.element.doi-10_2478_s11533-009-0046-z,
author = {Amine Fawaz},
title = {Total curvature and volume of foliations on the sphere S 2},
journal = {Open Mathematics},
volume = {7},
year = {2009},
pages = {660-669},
zbl = {1185.53026},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0046-z}
}
Amine Fawaz. Total curvature and volume of foliations on the sphere S 2. Open Mathematics, Tome 7 (2009) pp. 660-669. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0046-z/
[1] Aubin T., Some nonlinear problems in Riemannian geometry, Springer-Verlag, Berlin, 1998
[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] | Zbl 1057.53022
[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] Fathi A., Laudenbach F., Poenaru V., Travaux de Thurston sur les surfaces, Astérisque, 1979, vol. 66–67, SMF Paris (in French)
[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] | Zbl 1067.78005
[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] | Zbl 0989.53020
[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] | Zbl 1006.53053
[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] | Zbl 0605.53022
[9] Johnson L.D., Volumes of flows, Proc. Amer. Math. Soc., 1988, 104, 923–932 http://dx.doi.org/10.2307/2046818[Crossref] | Zbl 0687.58031
[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] | Zbl 0502.53026
[11] Pavlov V., Buisine D., Goncharov V., Formation of vortex clusters on a sphere, Nonlinear processes in geophysics, 2001, 8, 9–19
[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] Thurston W., Three-dimensional geometry and topology, 1, Princeton University Press, 1997 | Zbl 0873.57001
[14] Tondeur P., Geometry of foliations, Monogr. Math., 90, Birkhäuser, 1997 | Zbl 0905.53002