Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets

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 $\mathrm{\Omega }$ 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 $\mathrm{A}\mathrm{u}\mathrm{t}({\mathrm{\Omega }}^{\mathrm{\prime }})$-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 $Z\subset \mathrm{\Omega }$. More precisely, if $\check{\mathrm{\Gamma }}\subset \mathrm{A}\mathrm{u}\mathrm{t}(\mathrm{\Omega })$ is a torsion-free discrete subgroup leaving $Z$ invariant such that $Z\mathrm{/}\check{\mathrm{\Gamma }}$ is compact, we prove that $Z \subset \Omega$istotallygeodesic\mathrm{.}Inparticular,letting$\Gamma \subset{\rm Aut}(\Omega)$beatorsion-freecocompactlattice,and$\pi: \Omega \to \Omega/\Gamma =: X_\Gamma$betheuniformizationmap,asubvariety$Y \subset X_\Gamma$ must be totally geodesic whenever some (and hence any) irreducible component $Z$ of ${\pi }^{-1}(Y)$ is an algebraic subset of $\mathrm{\Omega }$. For cocompact lattices this yields a characterization of totally geodesic subsets of $X_{\mathrm{\Gamma }}$ 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.