A Schwarz-type formula for minimal surfaces in Euclidean space $ \mathbb{R}^{n}$

Luis J. Alı́as; Pablo Mira

doi:10.1016/S1631-073X(02)02280-X

A Schwarz-type formula for minimal surfaces in Euclidean space

ℝ^{n}

[Une formule de type Schwarz pour les surfaces minimales de l'espace euclidien

ℝ^{n}

]

Luis J. Alı́as ¹ ; Pablo Mira ²

¹ Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
² Departamento de Matemática Aplicada y Estadı́stica, Universidad Politécnica de Cartagena, 30203 Cartagena, Murcia, Spain

Comptes Rendus. Mathématique, Volume 334 (2002) no. 5, pp. 389-394

Abstract (VO)
Résumé

This paper introduces a complex representation for minimal surfaces in $ℝ^{n}$ , based on the Schwarz formula which solves the classical Björling problem for minimal surfaces in $ℝ^{3}$ . As an application, it is shown that a k-dimensional plane of $ℝ^{n}$ is a plane of symmetry of a minimal surface in $ℝ^{n}$ 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 $ℝ^{n}$ .

Cet article présente une représentation complexe des surfaces minimales de $ℝ^{n}$ , basée sur la formule de Schwarz qui résout le problème classique de Björling pour les surfaces minimales de $ℝ^{3}$ . Comme application, nous montrons qu'un plan de dimension k de $ℝ^{n}$ est un plan de symetrie d'une surface minimale de $ℝ^{n}$ 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.

Reçu le : 2001-12-11
Accepté le : 2002-01-14
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02280-X

Affiliations des auteurs :

Luis J. Alı́as ¹ ; Pablo Mira ²

¹ Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
² Departamento de Matemática Aplicada y Estadı́stica, Universidad Politécnica de Cartagena, 30203 Cartagena, Murcia, Spain

@article{CRMATH_2002__334_5_389_0,
     author = {Luis J. Al{\i}́as and Pablo Mira},
     title = {A {Schwarz-type} formula for minimal surfaces in {Euclidean} space $ \mathbb{R}^{n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {389--394},
     year = {2002},
     publisher = {Elsevier},
     volume = {334},
     number = {5},
     doi = {10.1016/S1631-073X(02)02280-X},
     language = {en},
}

TY  - JOUR
AU  - Luis J. Alı́as
AU  - Pablo Mira
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
DO  - 10.1016/S1631-073X(02)02280-X
LA  - en
ID  - CRMATH_2002__334_5_389_0
ER  -

%0 Journal Article
%A Luis J. Alı́as
%A Pablo Mira
%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
%R 10.1016/S1631-073X(02)02280-X
%G en
%F CRMATH_2002__334_5_389_0

Luis J. Alı́as; Pablo Mira. A Schwarz-type formula for minimal surfaces in Euclidean space $ \mathbb{R}^{n}$. Comptes Rendus. Mathématique, Volume 334 (2002) no. 5, pp. 389-394. doi: 10.1016/S1631-073X(02)02280-X

Bibliographie
Cité par

[1] L.J. Alı́as, P. Mira, The Björling problem for minimal surfaces in $ℝ^{n}$ and its applications (in preparation)

[2] S. Alves, R.M.B. Chaves, P. Simoes, Björling's problem for minimal surfaces in $ℝ^{4}$ (in preparation)

[3] E.G. Björling 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] S.S. Chern; R. Osserman Complete minimal surfaces in Euclidean n-space, J. Analyse Math., Volume 19 (1967), pp. 15-34

[5] U. Dierkes; S. Hildebrant; A. Küster; O. Wohlrab Minimal Surfaces I, Comprehensive Studies in Math., 295, Springer-Verlag, 1992

[6] D. Hoffman; R. Osserman The geometry of the generalized Gauss map, Mem. Amer. Math. Soc., Volume 28 (1980)

[7] D.S.P. Leung The reflection principle for minimal submanifolds of Riemannian symmetric spaces, J. Differential Geom., Volume 8 (1973), pp. 153-160

[8] J.C.C. Nitsche, Lectures on Minimal Surfaces, I, Cambridge University Press, Cambridge, 1989

[9] R. Osserman A Survey of Minimal Surfaces, Dover Publications, New York, 1986

[10] R. Osserman Global properties of minimal surfaces in E³ and Eⁿ, Ann. Math., Volume 80 (1964), pp. 340-364

[11] H.A. Schwarz Gesammelte Mathematische Abhandlungen, Springer-Verlag, Berlin, 1890

Cité par Sources :

Commentaires - Politique