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

2026/08/02 by Yun Gao
Mathematics · #math.CV

paper · pdf

arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

Motivated by the gap phenomenon for proper holomorphic maps between complex unit balls, this paper investigates smooth CR maps f from an open piece M of the Shilov boundary of the unit ball B^s into the Shilov boundary S_r',s' of a higher-rank Type I bounded symmetric domain Ω_r',s'. To the best of our knowledge, this is the first work to systematically establish a general gap phenomenon in the higher-rank setting. Our main result demonstrates that when the signature difference s'-r' falls into the interval k(s-1) ≤ s'-r' < (k+1)(s-1) for some integer k, the non-trivial component of f is constrained to a much smaller Type I boundary. Up to automorphisms of the domain and target spaces, f decomposes into the block-diagonal form f(z) = \beginpmatrix I_r'-k 0
0 ϕ(z) \endpmatrix, where the essential component ϕ: M → S_{k, s'-r'+k} is a smooth CR map into the corresponding Shilov boundary. We provide explicit constructions showing that these dimensional bounds are sharp. As an immediate corollary, in the initial gap regime s-1 ≤ s' - r' < 2s-2, the map f reduces to the standard linear embedding.

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
• 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