Rigidity of holomorphic maps between bounded symmetric domains preserving Shilov boundary (2508.05980v1)

Abstract: We introduce the concept of orthogonal structure on complex Grassmannians. Based on this structure, we define the notion of orthogonal mappings. This class of maps generalizes holomorphic maps between type I bounded symmetric domains that preserve Shilov boundary. By analyzing the geometric properties of these orthogonal mappings, we obtain rigidity results for such mappings. As an application, we establish a rigidity result for holomorphic maps from rank $1$ type I bounded symmetric domains $\Omega_{1,s}$ (complex balls $B^s$) to higher-rank domains $ \Omega_{r',s'}$ that preserve their corresponding Shilov boundaries, where $s\ge 2$ and $2\le r'\le s'$. More specifically, we show that (1): such maps are constant when $s'-r' < s-1$; (2) for $s-1 \le s'-r' < 2s-2$, these maps reduce to standard linear embeddings after normalization by applying automorphisms on both sides. Our results are optimal and generalize the well-known optimal bounds for proper mappings between rank $1$ cases and CR maps between Shilov boundaries of higher-rank type I bounded symmetric domains.