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 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.

**1.**H. Alexander, Proper holomorphic mappings in , 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 , 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, 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**

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