A representation for pseudoholomorphic surfaces in spheres
HTML articles powered by AMS MathViewer
- by M. Dajczer and Th. Vlachos PDF
- Proc. Amer. Math. Soc. 144 (2016), 3105-3113 Request permission
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
- João Lucas Marquês Barbosa, On minimal immersions of $S^{2}$ into $S^{2m}$, Trans. Amer. Math. Soc. 210 (1975), 75–106. MR 375166, DOI 10.1090/S0002-9947-1975-0375166-2
- Robert L. Bryant, Submanifolds and special structures on the octonians, J. Differential Geometry 17 (1982), no. 2, 185–232. MR 664494
- Eugenio Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry 1 (1967), 111–125. MR 233294
- Chi Cheng Chen, The generalized curvature ellipses and minimal surfaces, Bull. Inst. Math. Acad. Sinica 11 (1983), no. 3, 329–336. MR 726980
- 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
- Marcos Dajczer and Luis A. Florit, A class of austere submanifolds, Illinois J. Math. 45 (2001), no. 3, 735–755. MR 1879232
- 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, DOI 10.1007/BF01245181
- Marcos Dajczer and Theodoros Vlachos, A class of minimal submanifolds in spheres. Preprint.
- 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, DOI 10.1090/memo/0236
- Theodoros Vlachos, Minimal surfaces, Hopf differentials and the Ricci condition, Manuscripta Math. 126 (2008), no. 2, 201–230. MR 2403186, DOI 10.1007/s00229-008-0174-y
Additional Information
- M. Dajczer
- Affiliation: Instituto National de Mathemática Pura e Applicada – Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, Brazil
- MR Author ID: 54140
- Email: marcos@impa.br
- Th. Vlachos
- Affiliation: Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
- MR Author ID: 291296
- Email: tvlachos@uoi.gr
- Received by editor(s): August 20, 2015
- Published electronically: March 18, 2016
- Communicated by: Lei Ni
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3105-3113
- MSC (2010): Primary 53C42; Secondary 53A10
- DOI: https://doi.org/10.1090/proc/12989
- MathSciNet review: 3487240