We study general properties of holomorphic isometric embeddings of complex unit balls
into bounded symmetric
domains of rank
.
In the first part, we study holomorphic isometries from
to
with nonminimal
isometric constants
for any irreducible bounded symmetric domain
of rank
, where
denotes
the canonical Kähler–Einstein metric on any irreducible bounded symmetric domain
normalized so that minimal
disks of
are of constant
Gaussian curvature
.
In particular, results concerning the upper bound of the dimension of isometrically
embedded
in
and the structure of the images of such holomorphic isometries are obtained.
In the second part, we study holomorphic isometries from
to
for any irreducible bounded
symmetric domains
of rank equal to
with
, where
is an integer
such that
is
the minimal embedding (i.e., the first canonical embedding) of the compact dual Hermitian
symmetric space
of
.
We completely classify images of all holomorphic isometries from
to
for
, where
. In particular, for
we prove that any
holomorphic isometry from
to
extends to some
holomorphic isometry from
to
.
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