Second fundamental forms of holomorphic isometries of the Poincar disk into bounded symmetric domains and their boundary behavior along the unit circle

作     者：MOK Ngaiming NG Sui Chung 

作者机构：Department of MathematicsThe University of Hong Kong 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2009年第52卷第12期

页      面：2628-2646页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：supported by the Research Grants Council of Hong Kong China (Grant No. CERG 7018/03) 

主　　题：holomorphic isometry Poincar disk Siegel upper half-plane second fundamental form asymptotics 

摘      要：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 *** the case of a germ of holomorphic isometry f:(Δ,λ ds 2Δ ;0) → (Ω,ds 2Ω ;0) of the Poincar disk Δ into a bounded symmetric domain Ω C N in its Harish-Chandra realization and equipped with the Bergman metric,f extends to a proper holomorphic isometric embedding F:(Δ,λ ds 2Δ) → (Ω,ds 2Ω) and Graph(f) extends to an affine-algebraic variety V  C × C *** of F which are not totally geodesic have been *** arise primarily from the p-th root map ρ p:H → H p and a non-standard holomorphic embedding G from the upper half-plane to the Siegel upper half-plane H 3 of genus *** the current article on the one hand we examine second fundamental forms σ of these known examples,by computing explicitly σ *** the other hand we study on the theoretical side asymptotic properties of σ for arbitrary holomorphic isometries of the Poincar disk into *** 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)=t 2 u(τ),where u t=0 ≡ *** show that u must satisfy the first order differential equation u t | t=0 ≡ 0 on the real axis outside a finite number of points at which u is *** a by-product of our method of proof we show that any non-standard holomorphic isometric embedding of the Poincar disk into the polydisk must develop singularities along the boundary *** equation φuφt | t=0 ≡ 0 along the real axis for holomorphic isometries into polydisks distinguishes the latter maps from holomorphic isometries into Siegel upper half-planes arising from *** the end of the article we formulate characterization problems for holomorphic isometries suggested both by the theoretical and the computational results of the article.