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)

 
 

 

Rigidity of proper holomorphic mappings between equidimensional bounded symmetric domains


Author: Zhen-Han Tu
Journal: Proc. Amer. Math. Soc. 130 (2002), 1035-1042
MSC (2000): Primary 32H02, 32M15
DOI: https://doi.org/10.1090/S0002-9939-01-06383-3
Published electronically: October 1, 2001
MathSciNet review: 1873777
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that any proper holomorphic mapping between two equidimensional irreducible bounded symmetric domains with rank $\geq 2$ is a biholomorphism. The proof of the main result in this paper will be achieved by a differential-geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their Bergman metrics.


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

  • 1. H. Alexander, Proper holomorphic mappings in $\mbox{$\mathbb C$ }^n$, Indiana Univ. Math. J. 26(1977), 134-146. MR 54:10685
  • 2. E. Bedford and S. Bell, Proper self maps of weakly pseudoconvex domains, Math. Ann. 261(1982), 47-49. MR 84c:32026
  • 3. S. Bell, Proper holomorphic correspondences between circular domains, Comment. Math. Helv. 57(1982), 532-538. MR 84m:32032
  • 4. S. Bell, Algebraic mappings of circular domains in $\mbox{$\mathbb C$ }^n$, in Several Complex Variables (edited by J. Fornaess), Math Notes Vol. 38, Princeton University Press, 1993, 126-135. MR 94a:32040
  • 5. S. Bell and R. Narasimhan, Proper holomorphic mappings of complex spaces, in Several Complex Variables VI (Barth and Narasimhan, Eds), Encyclopaedia of Math. Sciences Vol. 69, Springer-Verlag, 1990, 1-38. MR 92m:32046
  • 6. K. Diederich and J.E. Fornæss, Proper holomorphic images of strictly pseudoconvex domains, Math. Ann. 259(1982), 279-286. MR 83g:32026
  • 7. F. Forstneric, Proper holomorphic mappings: A survey, in Several Complex Variables (edited by J. Fornaess), Math Notes Vol. 38, Princeton University Press, 1993, 297-363. MR 94a:32042
  • 8. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978. MR 80k:53081
  • 9. G.M. Henkin and R. Novikov, Proper mappings of classical domains, in Linear and Complex Analysis Problem Book, Lecture Notes in Math. Vol. 1043, Springer, Berlin, 1984, 625-627.
  • 10. N. Mok, Uniqueness theorems of Hermitian metric of seminegative curvature on quotients of bounded symmetric domains, Ann. of Math. 125(1987), 105-152. MR 88f:32076
  • 11. N. Mok, Uniqueness theorem of Kähler metrics of semipositive holomorphic bisectional curvature on compact Hermitian symmetric space, Math. Ann. 276(1987), 177-204. MR 88c:53063
  • 12. N. Mok, Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds, Series in Pure Math. Vol. 6, World Scientific, Singapore, 1989.MR 92d:32046
  • 13. N. Mok and I-H. Tsai, Rigidity of convex realizations of irreducible bounded symmetric domains of rank$\geq2$, J. Reine Angew. Math. 431(1992), 91-122. MR 93h:32046
  • 14. I.I. Pyatetskii-shapiro, Automorphic Functions and the Geometry of Classical Domains, Gordon and Breach, New York, 1969.
  • 15. Y.T. Siu, The complex analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. 112(1980), 73-111. MR 81j:53061
  • 16. Y.T. Siu, Strong rigidity of compact quotients of exceptional bounded symmetric domains, Duke Math. J. 48(1981), 857-871. MR 86h:32053
  • 17. I-H. Tsai, Rigidity of holomorphic maps between compact Hermitian symmetric spaces, J. Diff. Geom. 33(1991), 717-729. MR 92d:32047
  • 18. I-H. Tsai, Rigidity of proper holomorphic maps between symmetric domains, J. Diff. Geom. 37(1993), 123-160. MR 93m:32038
  • 19. Z.-H. Tu, Rigidity of proper holomorphic maps between bounded symmetric domains, Ph.D. Thesis, The University of Hong Kong, May 2000.
  • 20. A.E. Tumanov and G.M. Henkin, Local characterization of holomorphic automorphisms of classical domains, Dokl. Akad. Nauk SSSR 267(1982), 796-799. (Russian) MR 85b:32048
  • 21. J.A. Wolf, Fine structure of Hermitian symmetric spaces, in Geometry of Symmetric Spaces (Boothby-Weiss, eds), Marcel-Dekker, New York, 1972, 271-357. MR 53:8516

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32H02, 32M15

Retrieve articles in all journals with MSC (2000): 32H02, 32M15


Additional Information

Zhen-Han Tu
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Address at time of publication: Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, People’s Republic of China
Email: Tuzhenhan@yahoo.com

DOI: https://doi.org/10.1090/S0002-9939-01-06383-3
Keywords: Bounded symmetric domains, Hermitian symmetric manifolds, proper holomorphic mappings, rigidity, totally geodesic submanifolds
Received by editor(s): September 29, 2000
Published electronically: October 1, 2001
Communicated by: Steven R. Bell
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society