  The current article stems from our study on the asymptotic behavior of
holomorphic isometric embeddings of the Poincar\'e disk into bounded symmetric
domains. As a first result we prove that any holomorphic curve exiting the
boundary of a bounded symmetric domain $\Omega$ must necessarily be
asymptotically totally geodesic. Assuming otherwise we derive by the method of
rescaling a hypothetical holomorphic isometric embedding of the Poincar\'e disk
with ${\rm Aut}(\Omega')$-equivalent tangent spaces into a tube domain $\Omega'
\subset \Omega$ and derive a contradiction by means of the Poincar\'e-Lelong
equation. We deduce that equivariant holomorphic embeddings between bounded
symmetric domains must be totally geodesic. Furthermore, we solve a
uniformization problem on algebraic subsets $Z \subset \Omega$. More precisely,
if $\check \Gamma\subset {\rm Aut}(\Omega)$ is a torsion-free discrete subgroup
leaving $Z$ invariant such that $Z/\check \Gamma$ is compact, we prove that $Z
\subset \Omega$ is totally geodesic. In particular, letting $\Gamma \subset{\rm
Aut}(\Omega)$ be a torsion-free cocompact lattice, and $\pi: \Omega \to
\Omega/\Gamma =: X_\Gamma$ be the uniformization map, a subvariety $Y \subset
X_\Gamma$ must be totally geodesic whenever some (and hence any) irreducible
component $Z$ of $\pi^{-1}(Y)$ is an algebraic subset of $\Omega$. For
cocompact lattices this yields a characterization of totally geodesic subsets
of $X_\Gamma$ by means of bi-algebraicity without recourse to the celebrated
monodromy result of Andr\'e-Deligne on subvarieties of Shimura varieties, and
as such our proof applies to not necessarily arithmetic cocompact lattices.

Differential Geometry;Complex Variables