Proper mappings between indefinite hyperbolic spaces and type I classical domains

Huang, Xiaojun; Lu, Jin; Tang, Xiaomin; Xiao, Ming

doi:10.1090/tran/8618

Proper mappings between indefinite hyperbolic spaces and type I classical domains
HTML articles powered by AMS MathViewer

by Xiaojun Huang, Jin Lu, Xiaomin Tang and Ming Xiao PDF

Trans. Amer. Math. Soc. 375 (2022), 8465-8481 Request permission

Abstract:

In this paper, we first study a mapping problem between indefinite hyperbolic spaces by employing the work established earlier by the authors. In particular, we generalize certain theorems proved by Baouendi-Ebenfelt-Huang [Amer. J. Math. 133 (2011), pp. 1633–1661] and Ng [Michigan Math. J. 62 (2013), pp. 769–777; Int. Math. Res. Not. IMRN 2 (2015), pp. 291–324]. Then we use these results to prove a rigidity result for proper holomorphic mappings between type I classical domains, which confirms a conjecture formulated by Chan [Int. Math. Res. Not., doi.org/10.1093/imrn/rnaa373] after the work of Zaitsev-Kim [Math. Ann. 362 (2015), pp. 639-677], Kim [Proper holomorphic maps between bounded symmetric domains, Springer, Tokyo, 2015, pp. 207–219] and himself.

References

M. Salah Baouendi, Peter Ebenfelt, and Xiaojun Huang, Holomorphic mappings between hyperquadrics with small signature difference, Amer. J. Math. 133 (2011), no. 6, 1633–1661. MR 2863372, DOI 10.1353/ajm.2011.0044
M. S. Baouendi, Peter Ebenfelt, and Linda P. Rothschild, Transversality of holomorphic mappings between real hypersurfaces in different dimensions, Comm. Anal. Geom. 15 (2007), no. 3, 589–611. MR 2379806, DOI 10.4310/CAG.2007.v15.n3.a6
M. S. Baouendi and Xiaojun Huang, Super-rigidity for holomorphic mappings between hyperquadrics with positive signature, J. Differential Geom. 69 (2005), no. 2, 379–398. MR 2169869, DOI 10.4310/jdg/1121449110
S. T. Chan, Rigidity of proper holomorphic maps between type-$\mathrm {I}$ irreducible bounded symmetric domains, Int. Math. Res. Not., doi.org/10.1093/imrn/rnaa373.
John P. D’Angelo and Ming Xiao, Symmetries in CR complexity theory, Adv. Math. 313 (2017), 590–627. MR 3649233, DOI 10.1016/j.aim.2017.04.014
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.
Xiaojun Huang, On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions, J. Differential Geom. 51 (1999), no. 1, 13–33. MR 1703603
X. Huang, On a semi-rigidity property for holomorphic maps, Asian J. Math., 7 (2003), no. 4, 463–492. (A special issue dedicated to Y. T. Siu on the ocassion of his 60th birthday.)
Xiaojun Huang and Shanyu Ji, Mapping $\mathbf B^n$ into $\mathbf B^{2n-1}$, Invent. Math. 145 (2001), no. 2, 219–250. MR 1872546, DOI 10.1007/s002220100140
Xiaojun Huang, Jin Lu, Xiaomin Tang, and Ming Xiao, Boundary characterization of holomorphic isometric embeddings between indefinite hyperbolic spaces, Adv. Math. 374 (2020), 107388, 37. MR 4157588, DOI 10.1016/j.aim.2020.107388
Sung-Yeon Kim, Proper holomorphic maps between bounded symmetric domains, Complex analysis and geometry, Springer Proc. Math. Stat., vol. 144, Springer, Tokyo, 2015, pp. 207–219. MR 3446758, DOI 10.1007/978-4-431-55744-9_{1}5
Sung-Yeon Kim and Dmitri Zaitsev, Rigidity of CR maps between Shilov boundaries of bounded symmetric domains, Invent. Math. 193 (2013), no. 2, 409–437. MR 3090183, DOI 10.1007/s00222-012-0430-3
Sung-Yeon Kim and Dmitri Zaitsev, Rigidity of proper holomorphic maps between bounded symmetric domains, Math. Ann. 362 (2015), no. 1-2, 639–677. MR 3343893, DOI 10.1007/s00208-014-1143-2
Ngaiming Mok, Nonexistence of proper holomorphic maps between certain classical bounded symmetric domains, Chinese Ann. Math. Ser. B 29 (2008), no. 2, 135–146. MR 2392329, DOI 10.1007/s11401-007-0174-3
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
Sui-Chung Ng, Proper holomorphic mappings on flag domains of $\textrm {SU}(p,q)$ type on projective spaces, Michigan Math. J. 62 (2013), no. 4, 769–777. MR 3160541, DOI 10.1307/mmj/1387226164
Sui-Chung Ng, Holomorphic double fibration and the mapping problems of classical domains, Int. Math. Res. Not. IMRN 2 (2015), 291–324. MR 3340322, DOI 10.1093/imrn/rnt200
S. Ng, Z. Tu., and W. Yin, Proper holomorphic mappings among bounded symmetric domains and related Hartogs domains, preprint, available at: http://math.ecnu.edu.cn/~scng/papers/Proper_holomorphic_mappings_among_bounded_symmetric_domains_and_related_Hartogs_domains.pdf (2015).
Aeryeong Seo, New examples of proper holomorphic maps among symmetric domains, Michigan Math. J. 64 (2015), no. 2, 435–448. MR 3359034, DOI 10.1307/mmj/1434731932
Aeryeong Seo, Proper holomorphic polynomial maps between bounded symmetric domains of classical type, Proc. Amer. Math. Soc. 144 (2016), no. 2, 739–751. MR 3430850, DOI 10.1090/proc12755
I Hsun Tsai, Rigidity of proper holomorphic maps between symmetric domains, J. Differential Geom. 37 (1993), no. 1, 123–160. MR 1198602
Zhen-Han Tu, Rigidity of proper holomorphic mappings between equidimensional bounded symmetric domains, Proc. Amer. Math. Soc. 130 (2002), no. 4, 1035–1042. MR 1873777, DOI 10.1090/S0002-9939-01-06383-3
Zhen-Han Tu, Rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains, Math. Z. 240 (2002), no. 1, 13–35. MR 1906705, DOI 10.1007/s002090100353
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