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Article: On the asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domains
Title  On the asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domains 

Authors  
Keywords  53B25 53C35 53C55 Analytic Continuation Asymptotic Geodesy Bergman Metric Bounded Symmetric Domain Holomorphic Isometry Poincaré Disk Second Fundamental Form, Siegel Upper HalfPlane Symplectic Geometry 
Issue Date  2009 
Publisher  Elsevier BV. The Journal's web site is located at http://www.elsevier.com/wps/find/journaldescription.cws_home/708310/description#description 
Citation  Acta Mathematica Scientia, 2009, v. 29 n. 4, p. 881902 How to Cite? 
Abstract  In this article we study holomorphic isometries of the Poincare disk into bounded symmetric domains. Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometries up to normalizing constants with respect to the Bergman metric, showing in particular that the graph V0 of any germ of holomorphic isometry of the Poincaré disk Δ into an irreducible bounded symmetric domain Ω {double subset} ℂN in its HarishChandra realization must extend to an affinealgebraic subvariety V C ⊂ = ℂ×ℂN = ℂN+1, and that the irreducible component of V∩(Δ×Ω) containing V0 is the graph of a proper holomorphic isometric embedding F: Δ → Ω. In this article we study holomorphic isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δΔ. Starting with the structural equation for holomorphic isometries arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form a of the holomorphic isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ∥ρ∥ must vanish at a general boundary point either to the order 1 or to the order 1/2 called a holomorphic isometry of the first resp. second kind. We deal with special cases of nonstandard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper halfplane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper halfplanes. © 2009 Wuhan Institute of Physics and Mathematics. 
Persistent Identifier  http://hdl.handle.net/10722/156241 
ISSN  2015 Impact Factor: 0.557 2015 SCImago Journal Rankings: 0.615 
References 
DC Field  Value  Language 

dc.contributor.author  Mok, N  en_US 
dc.date.accessioned  20120808T08:40:59Z   
dc.date.available  20120808T08:40:59Z   
dc.date.issued  2009  en_US 
dc.identifier.citation  Acta Mathematica Scientia, 2009, v. 29 n. 4, p. 881902  en_US 
dc.identifier.issn  02529602  en_US 
dc.identifier.uri  http://hdl.handle.net/10722/156241   
dc.description.abstract  In this article we study holomorphic isometries of the Poincare disk into bounded symmetric domains. Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometries up to normalizing constants with respect to the Bergman metric, showing in particular that the graph V0 of any germ of holomorphic isometry of the Poincaré disk Δ into an irreducible bounded symmetric domain Ω {double subset} ℂN in its HarishChandra realization must extend to an affinealgebraic subvariety V C ⊂ = ℂ×ℂN = ℂN+1, and that the irreducible component of V∩(Δ×Ω) containing V0 is the graph of a proper holomorphic isometric embedding F: Δ → Ω. In this article we study holomorphic isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δΔ. Starting with the structural equation for holomorphic isometries arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form a of the holomorphic isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ∥ρ∥ must vanish at a general boundary point either to the order 1 or to the order 1/2 called a holomorphic isometry of the first resp. second kind. We deal with special cases of nonstandard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper halfplane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper halfplanes. © 2009 Wuhan Institute of Physics and Mathematics.  en_US 
dc.language  eng  en_US 
dc.publisher  Elsevier BV. The Journal's web site is located at http://www.elsevier.com/wps/find/journaldescription.cws_home/708310/description#description  en_US 
dc.relation.ispartof  Acta Mathematica Scientia  en_US 
dc.rights  This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.   
dc.subject  53B25  en_US 
dc.subject  53C35  en_US 
dc.subject  53C55  en_US 
dc.subject  Analytic Continuation  en_US 
dc.subject  Asymptotic Geodesy  en_US 
dc.subject  Bergman Metric  en_US 
dc.subject  Bounded Symmetric Domain  en_US 
dc.subject  Holomorphic Isometry  en_US 
dc.subject  Poincaré Disk  en_US 
dc.subject  Second Fundamental Form, Siegel Upper HalfPlane  en_US 
dc.subject  Symplectic Geometry  en_US 
dc.title  On the asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domains  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  postprint  en_US 
dc.identifier.doi  10.1016/S02529602(09)60076X  en_US 
dc.identifier.scopus  eid_2s2.065549128135  en_US 
dc.identifier.hkuros  158320   
dc.relation.references  http://www.scopus.com/mlt/select.url?eid=2s2.065549128135&selection=ref&src=s&origin=recordpage  en_US 
dc.identifier.volume  29  en_US 
dc.identifier.issue  4  en_US 
dc.identifier.spage  881  en_US 
dc.identifier.epage  902  en_US 
dc.publisher.place  Netherlands  en_US 
dc.identifier.scopusauthorid  Mok, N=7004348032  en_US 