Differential Geometry
On bounds for total absolute curvature of surfaces in hyperbolic 3-space
[Quelques bornes pour la courbure totale de surfaces dans l'espace hyperbolique de dimension 3]

Présenté par : Étienne Ghys

Langevin, Rémi1 ; Solanes, Gil2
1 Laboratoire de Topologie, Université de Bourgogne, UMR 5584 (CNRS) BP 47870, 21078 Dijon cedex, France
2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Comptes Rendus. Mathématique, Tome 336 (2003) no. 1, pp. 47-50.

Nous construisons des exemples de surfaces dans l'espace hyperbolique qui ne satisfont pas l'inégalité de Chern–Lashof (qui est vérifiée pour les surfaces immergées dans l'espace euclidien).

We construct examples of surfaces in hyperbolic space which do not satisfy the Chern–Lashof inequality (which holds for immersed surfaces in Euclidean space).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)00003-1
Langevin, Rémi 1 ; Solanes, Gil 2

1 Laboratoire de Topologie, Université de Bourgogne, UMR 5584 (CNRS) BP 47870, 21078 Dijon cedex, France
2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain 
@article{CRMATH_2003__336_1_47_0,
     author = {Langevin, R\'emi and Solanes, Gil},
     title = {On bounds for total absolute curvature of surfaces in hyperbolic 3-space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {47--50},
     publisher = {Elsevier},
     volume = {336},
     number = {1},
     year = {2003},
     doi = {10.1016/S1631-073X(02)00003-1},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)00003-1/}
}
TY  - JOUR
AU  - Langevin, Rémi
AU  - Solanes, Gil
TI  - On bounds for total absolute curvature of surfaces in hyperbolic 3-space
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 47
EP  - 50
VL  - 336
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(02)00003-1/
DO  - 10.1016/S1631-073X(02)00003-1
LA  - en
ID  - CRMATH_2003__336_1_47_0
ER  - 
%0 Journal Article
%A Langevin, Rémi
%A Solanes, Gil
%T On bounds for total absolute curvature of surfaces in hyperbolic 3-space
%J Comptes Rendus. Mathématique
%D 2003
%P 47-50
%V 336
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(02)00003-1/
%R 10.1016/S1631-073X(02)00003-1
%G en
%F CRMATH_2003__336_1_47_0
Langevin, Rémi; Solanes, Gil. On bounds for total absolute curvature of surfaces in hyperbolic 3-space. Comptes Rendus. Mathématique, Tome 336 (2003) no. 1, pp. 47-50. doi : 10.1016/S1631-073X(02)00003-1. http://www.numdam.org/articles/10.1016/S1631-073X(02)00003-1/

[1] Chern, S.S.; Lashof, R.K. On the total absolute curvature of immersed manifolds I, Amer. J. Math., Volume 79 (1957), pp. 306-318

[2] Langevin, R.; Rosenberg, H. Fenchel type theorems for submanifolds of 𝕊 n , Comment. Math. Helv., Volume 71 (1996), pp. 594-616

[3] Teufel, E. On the total absolute curvature of immersions into hyperbolic spaces, Topics in Differential Geometry, Vols. I, II, Colloq. Math. Soc. Janos Bolyai, 46, 1988, pp. 1201-1209

[4] Saleemi, B.A.; Willmore, T.J. Total absolute curvature of immersed manifolds, J. London Math. Soc., Volume 41 (1966), pp. 153-160

[5] Brickell, F.; Hsiung, C.C. The total absolute curvature of closed curves in Riemannian manifolds, J. Differential Geom., Volume 9 (1974), pp. 177-193

[6] Spivak, M. A comprehensive introduction to differential geometry, Publish or Perish, Wilmington, 1979

[7] Teufel, E. Differential topology and the computation of total absolute curvature, Math. Ann., Volume 258 (1982), pp. 471-480

[8] Bakelman, I.J. Convex Analysis and Nonlinear Geometric Elliptic Equations, Springer-Verlag, 1994

[9] Rockafellar, R.T. Convex Analysis, Princeton University Press, Princeton, NJ, 1970

Cité par Sources :