Uniformization problems on subvarieties of finite-volume quotient spaces of bounded symmetric domains

By the Uniformization Theorem a compact Riemann surface of genus $\ge 2$ is uniformized by the unit disk $\Delta$ and equivalently by the upper half plane $\mathcal H$. $\mathcal H$ is also the universal covering space of the moduli space of elliptic curves equipped with an appropriate level structure. In Several Complex Variables, the Siegel upper half plane $\mathcal H_g, g \ge 1$ is an analogue of $\mathcal H = \mathcal H_1$, and it is the universal covering space of moduli spaces of polarized Abelian varieties with appropriate level structures. $\mathcal H_g$ belongs, up to biholomorphic equivalence, to the set of bounded symmetric domains, on which a great deal of mathematical research is taking place. Especially, finite-volume quotients of bounded symmetric domains $\Omega$, which are naturally quasi-projective varieties, are objects of immense interest to Several Complex Variables, Algebraic Geometry, Arithmetic Geometry and Number Theory, and an important topic is the study of covering spaces of algebraic subsets of such quasi-projective varieties. While a lot has already been achieved in the case of Shimura varieties by means of methods of Diophantine Geometry, Model Theory, Hodge Theory and Complex Differential Geometry, techniques for the general case of not necessarily arithmetic quotients $\Omega/\Gamma$ have just begun to be developed. We will explain a differential-geometric approach leading to various characterization results for totally geodesic subvarieties of finite-volume quotients $\Omega/\Gamma$ without the assumption of arithmeticity. Especially, we will explain how the study of holomorphic isometric embeddings of the Poincaré disk and more generally complex unit balls into bounded symmetric domains can be further developed to derive uniformization theorems for bi-algebraic varieties and more generally for the Zariski closure of images of algebraic sets under the universal covering map.