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

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 f: (Δ, λ ds ²_Δ ;0) → (Ω, ds ²_Ω ;0) of the Poincaré 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 ²_Δ ;) → (Ω, ds ²_Ω ) and Graph(f) extends to an affine-algebraic variety V ⊂ ℂ × ℂ^N. Examples of F which are not totally geodesic have been constructed. They 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 3. In the current article on the one hand we examine second fundamental forms σ of these known examples, by computing explicitly φ = |σ|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) = t ² u(τ), where u| _t=0 ≢ 0. We show that u must satisfy the first order differential equation δu/δt| _t=0 ≡ 0 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 δu/δt| _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 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.