Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


A representation for pseudoholomorphic surfaces in spheres

Authors: M. Dajczer and Th. Vlachos
Journal: Proc. Amer. Math. Soc. 144 (2016), 3105-3113
MSC (2010): Primary 53C42; Secondary 53A10
Published electronically: March 18, 2016
MathSciNet review: 3487240
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a local representation for the pseudoholomorphic surfaces in Euclidean spheres in terms of holomorphic data. Similar to the case of the generalized Weierstrass representation of Hoffman and Osserman for minimal Euclidean surfaces, we assign such a surface in $ \mathbb{S}^{2n}$ to a given set of $ n$ holomorphic functions defined on a simply-connected domain in $ \mathbb{C}$.

References [Enhancements On Off] (What's this?)

  • [1] João Lucas Marquês Barbosa, On minimal immersions of $ S^{2}$ into $ S^{2m}$, Trans. Amer. Math. Soc. 210 (1975), 75-106. MR 0375166 (51 #11362)
  • [2] Robert L. Bryant, Submanifolds and special structures on the octonians, J. Differential Geom. 17 (1982), no. 2, 185-232. MR 664494 (84h:53091)
  • [3] Eugenio Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry 1 (1967), 111-125. MR 0233294 (38 #1616)
  • [4] Chi Cheng Chen, The generalized curvature ellipses and minimal surfaces, Bull. Inst. Math. Acad. Sinica 11 (1983), no. 3, 329-336. MR 726980 (85g:53008)
  • [5] Marcos Dajczer and Detlef Gromoll, Real Kaehler submanifolds and uniqueness of the Gauss map, J. Differential Geom. 22 (1985), no. 1, 13-28. MR 826421 (87g:53088b)
  • [6] Marcos Dajczer and Luis A. Florit, A class of austere submanifolds, Illinois J. Math. 45 (2001), no. 3, 735-755. MR 1879232 (2003g:53090)
  • [7] Marcos Dajczer and Detlef Gromoll, The Weierstrass representation for complete minimal real Kaehler submanifolds of codimension two, Invent. Math. 119 (1995), no. 2, 235-242. MR 1312499 (96c:53095),
  • [8] Marcos Dajczer and Theodoros Vlachos, A class of minimal submanifolds in spheres. Preprint.
  • [9] David A. Hoffman and Robert Osserman, The geometry of the generalized Gauss map, Mem. Amer. Math. Soc. 28 (1980), no. 236, iii+105. MR 587748 (82b:53012),
  • [10] Theodoros Vlachos, Minimal surfaces, Hopf differentials and the Ricci condition, Manuscripta Math. 126 (2008), no. 2, 201-230. MR 2403186 (2008m:53157),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C42, 53A10

Retrieve articles in all journals with MSC (2010): 53C42, 53A10

Additional Information

M. Dajczer
Affiliation: Instituto National de Mathemática Pura e Applicada – Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, Brazil

Th. Vlachos
Affiliation: Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece

Received by editor(s): August 20, 2015
Published electronically: March 18, 2016
Communicated by: Lei Ni
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society