Comptes Rendus
no. 5
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. Alı́as1 ; Pablo Mira2
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.

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.

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 .

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02280-X
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 
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     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},
     publisher = {Elsevier},
     volume = {334},
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     year = {2002},
     doi = {10.1016/S1631-073X(02)02280-X},
     language = {en},
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02280-X/

[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

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