Abstract:

In this article we study holomorphic isometries of the Poincar´e 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 normal-izing constants with respect to the Bergman metric, showing in particular that the graph V0 of any germ of holomorphic isometry of the Poincar´e disk Δ into an irreducible bounded symmetric domain  Ω   C^N in its Harish-Chandra realization must extend to an affine-algebraic subvariety V ⊂ C×C^N = C^N+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 ∈ ∂Δ. Starting with the structural equation for holomor-phic 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 σ of the holomorphic isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we
prove that ||σ|| 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
Poincaré 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 Poincar´e 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.

Key words: holomorphic isometry, Bergman metric, Poincar′e disk, analytic continua-tion, bounded symmetric domain, asymptotic geodesy, second fundamental form, Siegel upper half-plane, symplectic geometry