Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Pseudo-holomorphic curves in complex Grassmann manifolds


Authors: Xiaoxiang Jiao and Jiagui Peng
Journal: Trans. Amer. Math. Soc. 355 (2003), 3715-3726
MSC (2000): Primary 53C42, 53C55
DOI: https://doi.org/10.1090/S0002-9947-03-03244-6
Published electronically: May 7, 2003
MathSciNet review: 1990170
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that the Kähler angle of the pseudo-holomorphic sphere of constant curvature in complex Grassmannians is constant. At the same time we also prove several pinching theorems for the curvature and the Kähler angle of the pseudo-holomorphic spheres in complex Grassmannians with non-degenerate associated harmonic sequence.


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

  • 1. J. Bolton, G. R. Jensen, M. Rigoli, and L. M. Woodward, On conformal minimal immersions of ${\mathbf{S}}^2$ into ${\mathbf{CP}}^n$, Math. Ann., 279(1988), 599-620. MR 88m:53110
  • 2. E. Calabi, Isometric embedding of complex manifolds, Ann. Math. (2), 58(1953), 1-23. MR 15:160c
  • 3. S. S. Chern and J. G. Wolfson, Harmonic maps of the two-sphere into a complex Grassmann manifold, II, Ann. Math., 125(1987), 301-335. MR 88g:58038
  • 4. S. S. Chern and J. G. Wolfson, Minimal surfaces by moving frames, Amer. J. Math., 105(1983), 59-83. MR 84i:53056
  • 5. Q. Chi and Y. Zheng, Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifolds, Trans. Amer. Math. Soc., 313(1989), 393-406. MR 90m:53072
  • 6. P. Griffiths and J. Harris, Principles of algebraic geometry, Pure and Applied Mathematics, London, New York: Wiley, 1978. MR 80b:14001
  • 7. X. X. Jiao, On harmonic maps of surfaces into complex Grassmannians, Chinese Ann. Math., 21A(1)(2000), 57-60. MR 2001b:53083
  • 8. H. B. Lawson, The Riemannian geometry of holomorphic curves, Proc. Conf. Holomorphic Mapping and Minimal Surfaces, Bol. Soc. Brasil. Mat., 2(1971), 45-62. MR 48:2957
  • 9. K. Uhlenbeck, Harmonic maps into Lie groups (classical solutions of the chiral model), J. Differential Geom., 30(1989), 1-50. MR 90g:58028
  • 10. J. G. Wolfson, Harmonic sequences and harmonic maps of surfaces into complex Grassmann manifolds, J. Differential Geom., 27(1988), 161-178. MR 89c:58031
  • 11. Y. B. Zheng, Quantization of curvature of harmonic two-spheres in Grassmann manifolds, Trans. Amer. Math. Soc., 316(1)(1989), 193-214. MR 90b:58055
  • 12. K. Yang, Complete and compact minimal surfaces, Kluwer Academic Publishers, 1989. MR 91h:53058
  • 13. K. Yang, Compact Riemann surfaces and algebraic curves, Series in Pure Mathematics, Vol. 10, World Scientific, 1988. MR 90e:14023
  • 14. X. X. Jiao and J. G. Peng, A classification of holomorphic two-spheres with constant curvature in complex Grassmannians, Differential Geom. Appl., to appear.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C42, 53C55

Retrieve articles in all journals with MSC (2000): 53C42, 53C55


Additional Information

Xiaoxiang Jiao
Affiliation: Department of Mathematics, Graduate School, Chinese Academy of Sciences, Beijing 100039, China
Email: xxj@gscas.ac.cn

Jiagui Peng
Affiliation: Department of Mathematics, Graduate School, Chinese Academy of Sciences, Beijing 100039, China
Email: pengck@gscas.ac.cn

DOI: https://doi.org/10.1090/S0002-9947-03-03244-6
Keywords: Gauss curvature, K\"{a}hler angle, harmonic sequence, pseudo-holomorphic curve
Received by editor(s): September 6, 2002
Received by editor(s) in revised form: October 31, 2002
Published electronically: May 7, 2003
Additional Notes: Supported by the National Natural Science Foundation of China (Grants No. 10001033, 10131020, 10071804) and the President Foundation of the Graduate School of the Chinese Academy of Sciences
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society