AI 中文总结
本文研究更高秩I型有界对称域Shilov边界间光滑CR映射的间隙现象，建立了一般间隙定理，给出映射的块对角分解形式，证明维数界的紧性，并得到初始间隙区间上映射退化为标准线性嵌入的推论。
AI 中文摘要
受复单位球之间逆紧全纯映射间隙现象的启发，本文研究了从单位球$B^s$的Shilov边界的开片$M$到更高秩I型有界对称域$Ω_{r',s'}$的Shilov边界$S_{r',s'}$的光滑CR映射$f$。据我们所知，这是首个在更高秩情形下系统建立一般间隙现象的工作。我们的主要结果表明，当符号差$s'-r'$落在区间$k(s-1) \le s'-r' < (k+1)(s-1)$（其中$k$为某个整数）时，$f$的非平凡分量被约束在一个小得多的I型边界上。在定义域和目标空间的自同构意义下，$f$可分解为块对角形式$$f(z) = \begin{pmatrix} I_{r'-k} & 0 \\\\ 0 & ϕ(z) \end{pmatrix},$$其中本质分量$ϕ: M \to S_{k, s'-r'+k}$是到对应Shilov边界的光滑CR映射。我们给出了显式构造，证明这些维数界是紧的。作为直接推论，在初始间隙区间$s-1 \le s' - r' < 2s-2$中，映射$f$退化为标准线性嵌入。
英文摘要
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 $k(s-1) \le s'-r' < (k+1)(s-1)$ 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 $$f(z) = \begin{pmatrix} I_{r'-k} & 0 \\ 0 & ϕ(z) \end{pmatrix},$$ where the essential component $ϕ: M \to S_{k, s'-r'+k}$ 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 $s-1 \le s' - r' < 2s-2$, the map $f$ reduces to the standard linear embedding.
