Rigidity of holomorphic maps between Shilov boundaries of type-I bounded symmetric domains

2025/08/08 by Gao, Yun
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.05980

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 the Shilov boundaries of type-I bounded symmetric domains. 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 the holomorphic maps from the Shilov boundary of rank 1 type I bounded symmetric domain Ω_1,s (i.e. unit spheres) to the Shilov boundary of a higher-rank type-I domain Ω_r',s', where s≥ 2 and 2≤ r'≤ s'. More specifically, we show that: (1) such maps are constant when s'-r' < s-1; (2) for s-1 ≤ s'-r' < 2s-2, they reduce to the 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.

Citations

• Proper holomorphic maps between bounded symmetric domains with small rank differences
• Local orthogonal maps and rigidity of holomorphic mappings between real hyperquadrics
• Rigidity of proper holomorphic maps between type-I irreducible bounded symmetric domains
• Holomorphic maps between closed SU(l,m)-orbits in Grassmannian
• Holomorphic maps from the complex unit ball to Type IV classical domains
• New examples of proper holomorphic maps among symmetric domains
• Proper holomorphic polynomial maps between bounded symmetric domains of classical type
• On the Third Gap for Proper Holomorphic Maps between Balls

Related

• Rigidity of proper holomorphic maps between type-I irreducible bounded symmetric domains
• Proper holomorphic maps between bounded symmetric domains with small rank differences
• Holomorphic maps from the complex unit ball to Type IV classical domains
• Hermitian Rank and Rigidity of Holomorphic Mappings
• A rigidity result for proper holomorphic maps between balls