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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, thenup to automorphisms, $f$ is of the form $f=\imath\circ F$, where $F = F_1\timesF_2\colon \Omega\to \Omega_1'\times \Omega_2'$. Here $\Omega_1'$, $\Omega_2'$are bounded symmetric domains, the map $F_1\colon \Omega \to \Omega_1'$ is astandard embedding, $F_2: \Omega \to \Omega_2'$, and $\imath\colon\Omega'_1\times \Omega'_2 \to \Omega'$ is a totally geodesic holomorphicisometric embedding. Moreover we show that, under the rank condition above,there exists no proper holomorphic map $f: \Omega \to \Omega'$ if $\Omega$ isof 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 properholomorphic maps on maximal boundary components of $\Omega$, we constructrational maps between moduli spaces of subgrassmannians of compact duals of$\Omega$ and $\Omega'$, and induced CR-maps between CR-hypersurfaces of mixedsignature, thereby forcing the moduli map to satisfy strong localdifferential-geometric constraints (or that such moduli maps do not exist), andcomplete the proofs from rigidity results on geometric substructures modeled oncertain admissible pairs of rational homogeneous spaces of Picard number 1.

Proper holomorphic maps between bounded symmetric domains with small rank differences