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

TitleOn the asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domains
Authors
Keywords53B25
53C35
53C55
Analytic Continuation
Asymptotic Geodesy
Bergman Metric
Bounded Symmetric Domain
Holomorphic Isometry
Poincaré Disk
Second Fundamental Form, Siegel Upper Half-Plane
Symplectic Geometry
Issue Date2009
PublisherElsevier 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. 881-902 How to Cite?
AbstractIn 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 Harish-Chandra realization must extend to an affine-algebraic 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 non-standard 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 half-plane 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 half-planes. © 2009 Wuhan Institute of Physics and Mathematics.
Persistent Identifierhttp://hdl.handle.net/10722/156241
ISSN
2015 Impact Factor: 0.557
2015 SCImago Journal Rankings: 0.615
References

 

DC FieldValueLanguage
dc.contributor.authorMok, Nen_US
dc.date.accessioned2012-08-08T08:40:59Z-
dc.date.available2012-08-08T08:40:59Z-
dc.date.issued2009en_US
dc.identifier.citationActa Mathematica Scientia, 2009, v. 29 n. 4, p. 881-902en_US
dc.identifier.issn0252-9602en_US
dc.identifier.urihttp://hdl.handle.net/10722/156241-
dc.description.abstractIn 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 Harish-Chandra realization must extend to an affine-algebraic 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 non-standard 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 half-plane 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 half-planes. © 2009 Wuhan Institute of Physics and Mathematics.en_US
dc.languageengen_US
dc.publisherElsevier BV. The Journal's web site is located at http://www.elsevier.com/wps/find/journaldescription.cws_home/708310/description#descriptionen_US
dc.relation.ispartofActa Mathematica Scientiaen_US
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.subject53B25en_US
dc.subject53C35en_US
dc.subject53C55en_US
dc.subjectAnalytic Continuationen_US
dc.subjectAsymptotic Geodesyen_US
dc.subjectBergman Metricen_US
dc.subjectBounded Symmetric Domainen_US
dc.subjectHolomorphic Isometryen_US
dc.subjectPoincaré Disken_US
dc.subjectSecond Fundamental Form, Siegel Upper Half-Planeen_US
dc.subjectSymplectic Geometryen_US
dc.titleOn the asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domainsen_US
dc.typeArticleen_US
dc.identifier.emailMok, N:nmok@hkucc.hku.hken_US
dc.identifier.authorityMok, N=rp00763en_US
dc.description.naturepostprinten_US
dc.identifier.doi10.1016/S0252-9602(09)60076-Xen_US
dc.identifier.scopuseid_2-s2.0-65549128135en_US
dc.identifier.hkuros158320-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-65549128135&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume29en_US
dc.identifier.issue4en_US
dc.identifier.spage881en_US
dc.identifier.epage902en_US
dc.publisher.placeNetherlandsen_US
dc.identifier.scopusauthoridMok, N=7004348032en_US

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