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

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 $S_{r&#39;,s&#39;}$ of a higher-rank Type I bounded symmetric domain $Ω_{r&#39;,s&#39;}$. 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 $s&#39;-r&#39;$ falls into the interval $k(s-1) \le s&#39;-r&#39; < (k+1)(s-1)$ 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 $$f(z) = \begin{pmatrix} I_{r&#39;-k} & 0 \\ 0 & ϕ(z) \end{pmatrix},$$ where the essential component $ϕ: M \to S_{k, s&#39;-r&#39;+k}$ 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 $s-1 \le s&#39; - r&#39; < 2s-2$, the map $f$ reduces to the standard linear embedding.