Extension and rigidity problems for holomorphic isometries
into possibly infinite-dimensional space forms dated back
to works of Bochner and Calabi. For a bounded domain
D in C^n equipped with the Bergman kernel
K(z,w), the function log K_D(z,z) serves
as a potential function for the Bergman metric ds_D^2,
and the choice of an orthonormal basis for the Hilbert
space H^2(D) of square-integrable holomorphic functions
defines a holomorphic isometric embedding of
(D,ds_D^2) into the infinite-dimensional projective space
P^\infty equipped with the Fubini-Study metric. In the
simply connected case interior extension results already
follow from Calabi's seminal work in 1953 on the subject. Here
we are primarily concerned with extension beyond the
boundary for bounded domains with specific realizations,
notably bounded symmetric domains in their Harish-Chandra
realizations. The upshot is that the graph of a germ
of holomorphic isometry extends algebraically in the
latter case. On the other hand, we have found examples
of proper holomorphic isometric embeddings of the Poincar\'e
disk into bounded symmetric domains which are not totally
geodesic, giving in particular counter-examples to a
conjecture of Clozel-Ullmo's.
Bounded symmetric domains may be taken as prototypes, and generalizing the methods developed for these domains the problem of analytic continuation of holomorphic isometric immersions has now been solved in a very general setting. As an example, bounded symmetric domains share the common feature of admitting projective compactifications, given by the Borel embedding, such that the Bergman kernel K(z,w) extends as a rational function in z and the conjugate of w to the compactified space. We showed that the latter extension property is essentially what characterizes the Borel embedding as a `canonical' embedding up to finite coverings ramified outside the domain, and the same applies whenever the Bergman kernel K(z,w) is rational in z and the conjugate of w, in particular to bounded homogeneous domains realized as (unbounded) Siegel domains of the second kind.