Asymptotic behavior of holomorphic isometric embeddings of the Poincaré disk into bounded symmetric domains

Asymptotic behavior of holomorphic isometric embeddings
of the Poincaré disk into bounded symmetric domains

Prof. Ngaiming Mok ﹙莫毅明﹚
Department of Mathematics
The University of Hong Kong
Pokfulam, Hong Kong
nmok@hkucc.hku.hk

Abstract

Let f: (D, ds_D²;0) ® (W,g;0) be a germ of holomorphic isometric immersion of the Poincaré disk (D, ds_D²) into a bounded symmetric domain W equipped with a canonical Kähler-Einstein metric. It follows from the celebrated work of E. Calabi that f must extend to a global holomorphic isometry. We prove that Graph(f) extends algebraically, generalizing a result of Clozel-Ullmo in the special case where W is a polyidsk. We produced examples where f may fail to be totally geodesic.

In what follows we denote also by f: D®W the extended holomorphic isometric embedding. In this lecture we further study the asymptotic behavior of f. We show first of all that f must be asymptotically totally geodesic at a generic point of the boundary circle ¶D. We prove this by contradiction. Starting with the algebraic extension and dilating using automorphisms of the unit disk and W we obtain a hypothetic holomorphic isometric embedding F: D® W which resembles an equivariant embedding. (Among other things the the second fundamental form is nonzero and of the same norm everywhere.) We note that in the dual case Veronese embeddings give holomorphic isometric embeddings of the Riemann sphere into projective spaces, when both the domain and target spaces are equipped with Fubini-Study metrics, and in such cases the mapping is equivariant. The proof of the nonexistence of F: D® W requires therefore additional ingredients pertaining to the geometry of bounded symmetric domains. One of the elements of proof is the use of the Poincaré-Lelong equation and the Gauss-Bonnet Theorem which appeared in an earlier work of the author's on the characterization of totally geodesic compact holomorphic curves in Hermitian locally symmetric manifolds of the noncompact type.

We study further the asymptotic behavior of the second fundamental form. We show that if f: D® W is not totally geodesic, then it must be asymptotically geodesic of order [( 1 )/2] or 1. The counter-examples show precisely that both cases are possible. Furthermore, in the case of a holomorphic embedding of the unit disk D into the polydisk Dⁿ, we give a necessry and sufficient condition for f to be totally geodesic, by showing that this is the case if and only if the algebraic extension of Graph(f) does not develop any singularities on the boundary circle.

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On 22 May 2006, 11:33.