Proper holomorphic maps between bounded symmetric domains with small rank differences

In this paper we study the rigidity of proper holomorphic maps $f\colon \Omega\to\Omega'$ between irreducible bounded symmetric domains $\Omega$ and $\Omega'$ with small rank differences: $2\leq \text{rank}(\Omega')< 2\,\text{rank}(\Omega)-1$. More precisely, if either $\Omega$ and $\Omega'$ have the same type or $\Omega$ is of type~III and $\Omega'$ is of type~I, then up to automorphisms, $f$ is of the form $f=\imath\circ F$, where $F = F_1\times F_2\colon \Omega\to \Omega_1'\times \Omega_2'$. Here $\Omega_1'$, $\Omega_2'$ are bounded symmetric domains, the map $F_1\colon \Omega \to \Omega_1'$ is a standard embedding, $F_2: \Omega \to \Omega_2'$, and $\imath\colon \Omega'_1\times \Omega'_2 \to \Omega'$ is a totally geodesic holomorphic isometric embedding. Moreover we show that, under the rank condition above, there exists no proper holomorphic map $f: \Omega \to \Omega'$ if $\Omega$ is of type~I and $\Omega'$ is of type~III, or $\Omega$ is of type~II and $\Omega'$ is either of type~I or III. By considering boundary values of proper holomorphic maps on maximal boundary components of $\Omega$, we construct rational maps between moduli spaces of subgrassmannians of compact duals of $\Omega$ and $\Omega'$, and induced CR-maps between CR-hypersurfaces of mixed signature, thereby forcing the moduli map to satisfy strong local differential-geometric constraints (or that such moduli maps do not exist), and complete the proofs from rigidity results on geometric substructures modeled on certain admissible pairs of rational homogeneous spaces of Picard number 1.