Differential geometry
Sym–Bobenko formula for minimal surfaces in Heisenberg space
Comptes Rendus. Mathématique, Volume 351 (2013) no. 21-22, pp. 825-827.

We give an immersion formula, the Sym–Bobenko formula, for minimal surfaces in the 3-dimensional Heisenberg space. Such a formula can be used to give a generalized Weierstrass type representation and construct explicit examples of minimal surfaces.

On donne une formule dʼimmersions, dite de Sym–Bobenko, pour les surfaces minimales de lʼespace de Heisenberg de dimension 3. Une telle formule peut être utilisée pour écrire une représentation de Weierstrass généralisée et construire des exemples explicites de surfaces minimales.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.10.014
Cartier, Sébastien 1

1 Université Paris-Est, Laboratoire dʼanalyse et de mathématiques appliquées (UMR 8050), UPEC, UPEMLV, CNRS, 94010 Créteil, France
@article{CRMATH_2013__351_21-22_825_0,
     author = {Cartier, S\'ebastien},
     title = {Sym{\textendash}Bobenko formula for minimal surfaces in {Heisenberg} space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {825--827},
     publisher = {Elsevier},
     volume = {351},
     number = {21-22},
     year = {2013},
     doi = {10.1016/j.crma.2013.10.014},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.10.014/}
}
TY  - JOUR
AU  - Cartier, Sébastien
TI  - Sym–Bobenko formula for minimal surfaces in Heisenberg space
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 825
EP  - 827
VL  - 351
IS  - 21-22
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.10.014/
DO  - 10.1016/j.crma.2013.10.014
LA  - en
ID  - CRMATH_2013__351_21-22_825_0
ER  - 
%0 Journal Article
%A Cartier, Sébastien
%T Sym–Bobenko formula for minimal surfaces in Heisenberg space
%J Comptes Rendus. Mathématique
%D 2013
%P 825-827
%V 351
%N 21-22
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.10.014/
%R 10.1016/j.crma.2013.10.014
%G en
%F CRMATH_2013__351_21-22_825_0
Cartier, Sébastien. Sym–Bobenko formula for minimal surfaces in Heisenberg space. Comptes Rendus. Mathématique, Volume 351 (2013) no. 21-22, pp. 825-827. doi : 10.1016/j.crma.2013.10.014. http://www.numdam.org/articles/10.1016/j.crma.2013.10.014/

[1] Bobenko, A.I. All constant mean curvature tori in R3, S3, H3 in terms of theta functions, Math. Ann., Volume 290 (1991) no. 2, pp. 209-245

[2] Bobenko, A.I. Constant mean curvature surfaces and integrable equations, Russ. Math. Surv., Volume 46 (1991) no. 4, pp. 1-45

[3] Bobenko, A.I. Surfaces in terms of 2 by 2 matrices. Old and new integrable cases, Harmonic Maps and Integrable Systems, Aspects Math., vol. E23, Friedr. Vieweg, Braunschweig, 1994, pp. 83-127

[4] Brander, D.; Rossman, W.; Schmitt, N. Holomorphic representation of constant mean curvature surfaces in Minkowski space: consequences of non-compactness in loop group methods, Adv. Math., Volume 223 (2010), pp. 949-986

[5] Cartier, S. Surfaces des espaces homogènes de dimension 3, Université Paris-Est, 2011 (Ph.D. thesis)

[6] Daniel, B. The Gauss map of minimal surfaces in the Heisenberg group, Int. Math. Res. Not., Volume 3 (2011), pp. 674-695

[7] Dorfmeister, J.F.; Inoguchi, J.; Kobayashi, S. A loop group method for minimal surfaces in the three-dimensional Heisenberg group, 2012 (preprint) | arXiv

[8] Dorfmeister, J.F.; Pedit, F.; Wu, H. Weierstrass type representation of harmonic maps into symmetric spaces, Commun. Anal. Geom., Volume 6 (1998) no. 4, pp. 633-668

[9] Sym, A. Soliton surfaces and their applications (soliton geometry from spectral problems), Scheveningen, 1984 (Lect. Notes Phys.), Volume vol. 239, Springer, Berlin (1985), pp. 154-231

[10] Taniguchi, T. The Sym–Bobenko formula and constant mean curvature surfaces in Minkowski 3-space, Tokyo J. Math., Volume 20 (1997) no. 2, pp. 463-473

Cited by Sources:

This work is part of the authorʼs Ph.D. thesis [5].