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_{D}^{2};0) ® (W,g;0)
be a germ of holomorphic isometric immersion of the
Poincaré disk (D, ds_{D}^{2}) 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 D^{n}, 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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