[Une formule de type Schwarz pour les surfaces minimales de l'espace euclidien ]
Cet article présente une représentation complexe des surfaces minimales de , basée sur la formule de Schwarz qui résout le problème classique de Björling pour les surfaces minimales de . Comme application, nous montrons qu'un plan de dimension k de est un plan de symetrie d'une surface minimale de s'il lui est orthogonal. Nous décrivons aussi un procédé de construction de surfaces minimales ayant des propriétés géométriques prédéterminées, à partir de courbes analytiques réelles.
This paper introduces a complex representation for minimal surfaces in , based on the Schwarz formula which solves the classical Björling problem for minimal surfaces in . As an application, it is shown that a k-dimensional plane of is a plane of symmetry of a minimal surface in provided it intersects the surface orthogonally. A procedure for the construction of minimal surfaces is also described. This procedure introduces minimal surfaces with prescribed geometric properties, starting from real analytic curves in .
Accepté le :
Publié le :
@article{CRMATH_2002__334_5_389_0,
author = {Al{\i}́as, Luis J. and Mira, Pablo},
title = {A {Schwarz-type} formula for minimal surfaces in {Euclidean} space $ \mathbb{R}^{n}$},
journal = {Comptes Rendus. Math\'ematique},
pages = {389--394},
publisher = {Elsevier},
volume = {334},
number = {5},
year = {2002},
doi = {10.1016/S1631-073X(02)02280-X},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02280-X/}
}
TY - JOUR
AU - Alı́as, Luis J.
AU - Mira, Pablo
TI - A Schwarz-type formula for minimal surfaces in Euclidean space $ \mathbb{R}^{n}$
JO - Comptes Rendus. Mathématique
PY - 2002
SP - 389
EP - 394
VL - 334
IS - 5
PB - Elsevier
UR - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02280-X/
DO - 10.1016/S1631-073X(02)02280-X
LA - en
ID - CRMATH_2002__334_5_389_0
ER -
%0 Journal Article
%A Alı́as, Luis J.
%A Mira, Pablo
%T A Schwarz-type formula for minimal surfaces in Euclidean space $ \mathbb{R}^{n}$
%J Comptes Rendus. Mathématique
%D 2002
%P 389-394
%V 334
%N 5
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02280-X/
%R 10.1016/S1631-073X(02)02280-X
%G en
%F CRMATH_2002__334_5_389_0
Alı́as, Luis J.; Mira, Pablo. A Schwarz-type formula for minimal surfaces in Euclidean space $ \mathbb{R}^{n}$. Comptes Rendus. Mathématique, Tome 334 (2002) no. 5, pp. 389-394. doi : 10.1016/S1631-073X(02)02280-X. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02280-X/
[1] L.J. Alı́as, P. Mira, The Björling problem for minimal surfaces in and its applications (in preparation)
[2] S. Alves, R.M.B. Chaves, P. Simoes, Björling's problem for minimal surfaces in (in preparation)
[3] In integrationem aequationis derivatarum partialum superfici, cujus in puncto unoquoque principales ambo radii curvedinis aequales sunt sngoque contrario, Arch. Math. Phys., Volume 4 (1844) no. 1, pp. 290-315
[4] Complete minimal surfaces in Euclidean n-space, J. Analyse Math., Volume 19 (1967), pp. 15-34
[5] Minimal Surfaces I, Comprehensive Studies in Math., 295, Springer-Verlag, 1992
[6] The geometry of the generalized Gauss map, Mem. Amer. Math. Soc., Volume 28 (1980)
[7] The reflection principle for minimal submanifolds of Riemannian symmetric spaces, J. Differential Geom., Volume 8 (1973), pp. 153-160
[8] , Lectures on Minimal Surfaces, I, Cambridge University Press, Cambridge, 1989
[9] A Survey of Minimal Surfaces, Dover Publications, New York, 1986
[10] Global properties of minimal surfaces in E3 and En, Ann. Math., Volume 80 (1964), pp. 340-364
[11] Gesammelte Mathematische Abhandlungen, Springer-Verlag, Berlin, 1890
Cité par Sources :
