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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
Title | Second fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circle | ||||
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Authors | |||||
Keywords | Asymptotics Holomorphic Isometry Poincaré Disk Second Fundamental Form Siegel Upper Half-Plane | ||||
Issue Date | 2009 | ||||
Publisher | Science 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? | ||||
Abstract | Motivated 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 Identifier | http://hdl.handle.net/10722/156247 | ||||
ISSN | 2011 Impact Factor: 0.701 2012 SCImago Journal Rankings: 0.540 | ||||
ISI Accession Number ID |
Funding Information: This work was supported by the Research Grants Council of Hong Kong, China (Grant No. CERG 7018/03) | ||||
References |
DC Field | Value | Language |
---|---|---|
dc.contributor.author | Mok, N | en_US |
dc.contributor.author | Ng, SC | en_US |
dc.date.accessioned | 2012-08-08T08:41:01Z | - |
dc.date.available | 2012-08-08T08:41:01Z | - |
dc.date.issued | 2009 | en_US |
dc.identifier.citation | Science In China, Series A: Mathematics, 2009, v. 52 n. 12, p. 2628-2646 | en_US |
dc.identifier.issn | 1006-9283 | en_US |
dc.identifier.uri | http://hdl.handle.net/10722/156247 | - |
dc.description.abstract | Motivated 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.language | eng | en_US |
dc.publisher | Science in China Press. The Journal's web site is located at http://www.scichina.com:8081/sciAe/EN/volumn/current.shtml | en_US |
dc.relation.ispartof | Science in China, Series A: Mathematics | en_US |
dc.subject | Asymptotics | en_US |
dc.subject | Holomorphic Isometry | en_US |
dc.subject | Poincaré Disk | en_US |
dc.subject | Second Fundamental Form | en_US |
dc.subject | Siegel Upper Half-Plane | en_US |
dc.title | Second fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circle | en_US |
dc.type | Article | en_US |
dc.identifier.email | Mok, N:nmok@hkucc.hku.hk | en_US |
dc.identifier.authority | Mok, N=rp00763 | en_US |
dc.description.nature | link_to_subscribed_fulltext | en_US |
dc.identifier.doi | 10.1007/s11425-009-0222-4 | en_US |
dc.identifier.scopus | eid_2-s2.0-73249135709 | en_US |
dc.identifier.hkuros | 168371 | - |
dc.relation.references | http://www.scopus.com/mlt/select.url?eid=2-s2.0-73249135709&selection=ref&src=s&origin=recordpage | en_US |
dc.identifier.volume | 52 | en_US |
dc.identifier.issue | 12 | en_US |
dc.identifier.spage | 2628 | en_US |
dc.identifier.epage | 2646 | en_US |
dc.identifier.isi | WOS:000272622200007 | - |
dc.publisher.place | China | en_US |
dc.identifier.scopusauthorid | Mok, N=7004348032 | en_US |
dc.identifier.scopusauthorid | Ng, SC=35264831700 | en_US |
dc.identifier.citeulike | 6478253 | - |