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Article: Factorization of proper holomorphic maps on irreducible bounded symmetric domains of rank ≥ 2

TitleFactorization of proper holomorphic maps on irreducible bounded symmetric domains of rank ≥ 2
Authors
KeywordsBounded symmetric domain
Correspondence
Discriminant
Fatou's theorem
G-structure
Proper holomorphic map
Issue Date2010
PublisherScience China Press, co-published with Springer. The Journal's web site is located at http://math.scichina.com/english/
Citation
Science China Mathematics, 2010, v. 53 n. 3, p. 813-826 How to Cite?
AbstractWe obtain rigidity results on arbitrary proper holomorphic maps F from an irreducible bounded symmetric domain Ω of rank ≥ 2 into any complex space Z. After lifting to the normalization of the subvariety F(Ω) ⊂ Z, we prove that F must be the canonical projection map to the quotient space of Ω by a finite group of automorphisms. The approach is along the line of the works of Mok and Tsai by considering radial limits of bounded holomorphic functions derived from F and proving that proper holomorphic maps between bounded symmetric domains preserve certain totally geodesic subdomains. In contrast to the previous works, in general we have to deal with multivalent holomorphic maps for which Fatou's theorem cannot be applied directly. We bypass the difficulty by devising a limiting process for taking radial limits of correspondences arising from proper holomorphic maps and by elementary estimates allowing us to define distinct univalent branches of the underlying multivalent map on certain subsets. As a consequence of our rigidity result, with the exception of Type-IV domains, any proper holomorphic map f: Ω → D of Ω onto a bounded convex domain D is necessarily a biholomorphism. In the exceptional case where Ω is a Type-IV domain, either f is a biholomorphism or it is a double cover branched over a totally geodesic submanifold which can be explicitly described. © Science China Press and Springer-Verlag Berlin Heidelberg 2010.
Persistent Identifierhttp://hdl.handle.net/10722/75302
ISSN
2015 Impact Factor: 0.761
2015 SCImago Journal Rankings: 0.894
ISI Accession Number ID
Funding AgencyGrant Number
HKRGC, Hong KongGRF 7032/08P
National Natural Science Foundation of China10971156
Funding Information:

This work was partially supported by the GRF 7032/08P of the HKRGC, Hong Kong and National Natural Science Foundation of China (Grant No. 10971156). The first author wishes to thank the organizers of the International Conference on Complex Analysis and Related Topics, held August 2009 at the Chinese Academy of Sciences, Beijing, especially Professor Wang Yuefei, for their kind invitation and for the opportunity to take part in the memorable event in honor of Professor Yang Lo. The authors wish to dedicate this article to Professor Yang Lo on the occasion of his 70th birthday.

References

 

DC FieldValueLanguage
dc.contributor.authorMok, Nen_HK
dc.contributor.authorNg, SCen_HK
dc.contributor.authorTu, ZHen_HK
dc.date.accessioned2010-09-06T07:09:50Z-
dc.date.available2010-09-06T07:09:50Z-
dc.date.issued2010en_HK
dc.identifier.citationScience China Mathematics, 2010, v. 53 n. 3, p. 813-826en_HK
dc.identifier.issn1674-7283en_HK
dc.identifier.urihttp://hdl.handle.net/10722/75302-
dc.description.abstractWe obtain rigidity results on arbitrary proper holomorphic maps F from an irreducible bounded symmetric domain Ω of rank ≥ 2 into any complex space Z. After lifting to the normalization of the subvariety F(Ω) ⊂ Z, we prove that F must be the canonical projection map to the quotient space of Ω by a finite group of automorphisms. The approach is along the line of the works of Mok and Tsai by considering radial limits of bounded holomorphic functions derived from F and proving that proper holomorphic maps between bounded symmetric domains preserve certain totally geodesic subdomains. In contrast to the previous works, in general we have to deal with multivalent holomorphic maps for which Fatou's theorem cannot be applied directly. We bypass the difficulty by devising a limiting process for taking radial limits of correspondences arising from proper holomorphic maps and by elementary estimates allowing us to define distinct univalent branches of the underlying multivalent map on certain subsets. As a consequence of our rigidity result, with the exception of Type-IV domains, any proper holomorphic map f: Ω → D of Ω onto a bounded convex domain D is necessarily a biholomorphism. In the exceptional case where Ω is a Type-IV domain, either f is a biholomorphism or it is a double cover branched over a totally geodesic submanifold which can be explicitly described. © Science China Press and Springer-Verlag Berlin Heidelberg 2010.en_HK
dc.languageengen_HK
dc.publisherScience China Press, co-published with Springer. The Journal's web site is located at http://math.scichina.com/english/en_HK
dc.relation.ispartofScience China Mathematicsen_HK
dc.subjectBounded symmetric domainen_HK
dc.subjectCorrespondenceen_HK
dc.subjectDiscriminanten_HK
dc.subjectFatou's theoremen_HK
dc.subjectG-structureen_HK
dc.subjectProper holomorphic mapen_HK
dc.titleFactorization of proper holomorphic maps on irreducible bounded symmetric domains of rank ≥ 2en_HK
dc.typeArticleen_HK
dc.identifier.emailMok, N:nmok@hkucc.hku.hken_HK
dc.identifier.authorityMok, N=rp00763en_HK
dc.description.naturelink_to_OA_fulltext-
dc.identifier.doi10.1007/s11425-010-0058-yen_HK
dc.identifier.scopuseid_2-s2.0-77952175839en_HK
dc.identifier.hkuros169796en_HK
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-77952175839&selection=ref&src=s&origin=recordpageen_HK
dc.identifier.volume53en_HK
dc.identifier.issue3en_HK
dc.identifier.spage813en_HK
dc.identifier.epage826en_HK
dc.identifier.isiWOS:000276597700028-
dc.publisher.placeChinaen_HK
dc.identifier.scopusauthoridMok, N=7004348032en_HK
dc.identifier.scopusauthoridNg, SC=35264831700en_HK
dc.identifier.scopusauthoridTu, ZH=13702842300en_HK

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