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Article: Second fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circle

TitleSecond fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circle
Authors
KeywordsAsymptotics
Holomorphic Isometry
Poincaré Disk
Second Fundamental Form
Siegel Upper Half-Plane
Issue Date2009
PublisherScience in China Press. The Journal's web site is located at http://www.scichina.com:8081/sciAe/EN/volumn/current.shtml
Citation
Science In China, Series A: Mathematics, 2009, v. 52 n. 12, p. 2628-2646 How to Cite?
AbstractMotivated by problems arising from Arithmetic Geometry, in an earlier article one of the authors studied germs of holomorphic isometries between bounded domains with respect to the Bergman metric. In the case of a germ of holomorphic isometry, of the Poincaré disk Δ into a bounded symmetric domain Ω {double subset} ℂN in its Harish-Chandra realization and equipped with the Bergman metric, f extends to a proper holomorphic isometric embedding, and Graph(f) extends to an affine-algebraic variety V ⊂ ℂ × ℂN. Examples of F which are not totally geodesic have been constructed. They arise primarily from the p-th root map ρp: H → Hp and a non-standard holomorphic embedding G from the upper half-plane to the Siegel upper half-plane H3 of genus 3. In the current article on the one hand we examine second fundamental forms σ of these known examples, by computing explicitly φ = {pipe}{pipe}σ{pipe}{pipe}2. On the other hand we study on the theoretical side asymptotic properties of σ for arbitrary holomorphic isometries of the Poincaré disk into polydisks. For such mappings expressing via the inverse Cayley transform in terms of the Euclidean coordinate τ = s + it on the upper half-plane H, we have φ(τ) = t2u(τ), where u{pipe}t=0 ≢ 0. We show that u must satisfy the first order differential equation, on the real axis outside a finite number of points at which u is singular. As a by-product of our method of proof we show that any non-standard holomorphic isometric embedding of the Poincaré disk into the polydisk must develop singularities along the boundary circle. The equation, along the real axis for holomorphic isometries into polydisks distinguishes the latter maps from holomorphic isometries into Siegel upper half-planes arising from G. Towards the end of the article we formulate characterization problems for holomorphic isometries suggested both by the theoretical and the computational results of the article. © 2009 Science in China Press and Springer Berlin Heidelberg.
Persistent Identifierhttp://hdl.handle.net/10722/156247
ISSN
2011 Impact Factor: 0.701
2012 SCImago Journal Rankings: 0.540
ISI Accession Number ID
Funding AgencyGrant Number
Research Grants Council of Hong Kong, ChinaCERG 7018/03
Funding Information:

This work was supported by the Research Grants Council of Hong Kong, China (Grant No. CERG 7018/03)

References

 

DC FieldValueLanguage
dc.contributor.authorMok, Nen_US
dc.contributor.authorNg, SCen_US
dc.date.accessioned2012-08-08T08:41:01Z-
dc.date.available2012-08-08T08:41:01Z-
dc.date.issued2009en_US
dc.identifier.citationScience In China, Series A: Mathematics, 2009, v. 52 n. 12, p. 2628-2646en_US
dc.identifier.issn1006-9283en_US
dc.identifier.urihttp://hdl.handle.net/10722/156247-
dc.description.abstractMotivated by problems arising from Arithmetic Geometry, in an earlier article one of the authors studied germs of holomorphic isometries between bounded domains with respect to the Bergman metric. In the case of a germ of holomorphic isometry, of the Poincaré disk Δ into a bounded symmetric domain Ω {double subset} ℂN in its Harish-Chandra realization and equipped with the Bergman metric, f extends to a proper holomorphic isometric embedding, and Graph(f) extends to an affine-algebraic variety V ⊂ ℂ × ℂN. Examples of F which are not totally geodesic have been constructed. They arise primarily from the p-th root map ρp: H → Hp and a non-standard holomorphic embedding G from the upper half-plane to the Siegel upper half-plane H3 of genus 3. In the current article on the one hand we examine second fundamental forms σ of these known examples, by computing explicitly φ = {pipe}{pipe}σ{pipe}{pipe}2. On the other hand we study on the theoretical side asymptotic properties of σ for arbitrary holomorphic isometries of the Poincaré disk into polydisks. For such mappings expressing via the inverse Cayley transform in terms of the Euclidean coordinate τ = s + it on the upper half-plane H, we have φ(τ) = t2u(τ), where u{pipe}t=0 ≢ 0. We show that u must satisfy the first order differential equation, on the real axis outside a finite number of points at which u is singular. As a by-product of our method of proof we show that any non-standard holomorphic isometric embedding of the Poincaré disk into the polydisk must develop singularities along the boundary circle. The equation, along the real axis for holomorphic isometries into polydisks distinguishes the latter maps from holomorphic isometries into Siegel upper half-planes arising from G. Towards the end of the article we formulate characterization problems for holomorphic isometries suggested both by the theoretical and the computational results of the article. © 2009 Science in China Press and Springer Berlin Heidelberg.en_US
dc.languageengen_US
dc.publisherScience in China Press. The Journal's web site is located at http://www.scichina.com:8081/sciAe/EN/volumn/current.shtmlen_US
dc.relation.ispartofScience in China, Series A: Mathematicsen_US
dc.subjectAsymptoticsen_US
dc.subjectHolomorphic Isometryen_US
dc.subjectPoincaré Disken_US
dc.subjectSecond Fundamental Formen_US
dc.subjectSiegel Upper Half-Planeen_US
dc.titleSecond fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circleen_US
dc.typeArticleen_US
dc.identifier.emailMok, N:nmok@hkucc.hku.hken_US
dc.identifier.authorityMok, N=rp00763en_US
dc.description.naturelink_to_subscribed_fulltexten_US
dc.identifier.doi10.1007/s11425-009-0222-4en_US
dc.identifier.scopuseid_2-s2.0-73249135709en_US
dc.identifier.hkuros168371-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-73249135709&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume52en_US
dc.identifier.issue12en_US
dc.identifier.spage2628en_US
dc.identifier.epage2646en_US
dc.identifier.isiWOS:000272622200007-
dc.publisher.placeChinaen_US
dc.identifier.scopusauthoridMok, N=7004348032en_US
dc.identifier.scopusauthoridNg, SC=35264831700en_US
dc.identifier.citeulike6478253-

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