## Rigidity of proper holomorphic mappings between equidimensional bounded symmetric domains

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- by Zhen-Han Tu
- Proc. Amer. Math. Soc.
**130**(2002), 1035-1042 - DOI: https://doi.org/10.1090/S0002-9939-01-06383-3
- Published electronically: October 1, 2001
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## 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

- H. Alexander,
*Proper holomorphic mappings in $C^{n}$*, Indiana Univ. Math. J.**26**(1977), no. 1, 137–146. MR**422699**, DOI 10.1512/iumj.1977.26.26010 - Eric Bedford and Steve Bell,
*Proper self-maps of weakly pseudoconvex domains*, Math. Ann.**261**(1982), no. 1, 47–49. MR**675205**, DOI 10.1007/BF01456408 - Steven R. Bell,
*Proper holomorphic mappings between circular domains*, Comment. Math. Helv.**57**(1982), no. 4, 532–538. MR**694605**, DOI 10.1007/BF02565875 - S. Bell,
*Algebraic mappings of circular domains in $\mathbf C^n$*, Several complex variables (Stockholm, 1987/1988) Math. Notes, vol. 38, Princeton Univ. Press, Princeton, NJ, 1993, pp. 126–135. MR**1207857** - Steven R. Bell and Raghavan Narasimhan,
*Proper holomorphic mappings of complex spaces*, Several complex variables, VI, Encyclopaedia Math. Sci., vol. 69, Springer, Berlin, 1990, pp. 1–38. MR**1095089** - Klas Diederich and John E. Fornæss,
*Proper holomorphic images of strictly pseudoconvex domains*, Math. Ann.**259**(1982), no. 2, 279–286. MR**656667**, DOI 10.1007/BF01457314 - Franc Forstnerič,
*Proper holomorphic mappings: a survey*, Several complex variables (Stockholm, 1987/1988) Math. Notes, vol. 38, Princeton Univ. Press, Princeton, NJ, 1993, pp. 297–363. MR**1207867** - Sigurdur Helgason,
*Differential geometry, Lie groups, and symmetric spaces*, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**514561** - 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.
- Ngaiming Mok,
*Uniqueness theorems of Hermitian metrics of seminegative curvature on quotients of bounded symmetric domains*, Ann. of Math. (2)**125**(1987), no. 1, 105–152. MR**873379**, DOI 10.2307/1971290 - Ngaiming Mok,
*Uniqueness theorems of Kähler metrics of semipositive bisectional curvature on compact Hermitian symmetric spaces*, Math. Ann.**276**(1987), no. 2, 177–204. MR**870961**, DOI 10.1007/BF01450737 - Ngaiming Mok,
*Metric rigidity theorems on Hermitian locally symmetric manifolds*, Series in Pure Mathematics, vol. 6, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. MR**1081948**, DOI 10.1142/0773 - Ngaiming Mok and I Hsun Tsai,
*Rigidity of convex realizations of irreducible bounded symmetric domains of rank $\geq 2$*, J. Reine Angew. Math.**431**(1992), 91–122. MR**1179334** - I.I. Pyatetskii-shapiro, Automorphic Functions and the Geometry of Classical Domains, Gordon and Breach, New York, 1969.
- Yum Tong Siu,
*The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds*, Ann. of Math. (2)**112**(1980), no. 1, 73–111. MR**584075**, DOI 10.2307/1971321 - Yum Tong Siu,
*Strong rigidity of compact quotients of exceptional bounded symmetric domains*, Duke Math. J.**48**(1981), no. 4, 857–871. MR**782581**, DOI 10.1215/S0012-7094-81-04847-X - I Hsun Tsai,
*Rigidity of holomorphic maps between compact Hermitian symmetric spaces*, J. Differential Geom.**33**(1991), no. 3, 717–729. MR**1100208** - I Hsun Tsai,
*Rigidity of proper holomorphic maps between symmetric domains*, J. Differential Geom.**37**(1993), no. 1, 123–160. MR**1198602** - Z.-H. Tu, Rigidity of proper holomorphic maps between bounded symmetric domains, Ph.D. Thesis, The University of Hong Kong, May 2000.
- A. E. Tumanov and G. M. Khenkin,
*Local characterization of analytic automorphisms of classical domains*, Dokl. Akad. Nauk SSSR**267**(1982), no. 4, 796–799 (Russian). MR**681032** - Joseph A. Wolf,
*Fine structure of Hermitian symmetric spaces*, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Pure and Appl. Math., Vol. 8, Dekker, New York, 1972, pp. 271–357. MR**0404716**

## Bibliographic 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
- Received by editor(s): September 29, 2000
- Published electronically: October 1, 2001
- Communicated by: Steven R. Bell
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1873777