Shilov boundaries determine irreducible bounded symmetric domains
英文摘要:
We prove that irreducible symmetric domains are uniquely determined by the homotopy equivalence classes of their Shilov boundaries.
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Alexandru Chirvasitu
We prove that irreducible symmetric domains are uniquely determined by the homotopy equivalence classes of their Shilov boundaries.
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继续探索
We continue to discuss the example presented in \cite{JarPfl2015}. In particular, we clarify some gaps and complete the description of the Shilov boundary.
We give a counterexample to the following theorem of Bremermann on Shilov boundaries: if $D$ is a bounded domain in $\mathbb C^n$ having a univalent envelope of holomorphy, say $\widetilde D$, then the Shilov boundary of $D$ with respect to the algebra $\mathcal A(D)$ coincides with the corresponding one for $\widetilde D$.
Our goal is to establish what seems to be the first rigidity result for CR embeddings between Shilov boundaries of bounded symmetric domains of higher rank. The result states that any such CR embedding is the standard linear embedding up to CR automorphisms. Our basic assumption extends precisely the well-known optimal bound for the rank one case. There are no other restrictions on the ranks, in particular, the difficult case when the target rank is larger than the source rank is also allowed.
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Boundary Rigidity of Line-Bundle Shilov--Poisson Transforms on Bounded Symmetric Domains
研究有界对称域上线丛Shilov--Poisson变换的边界刚性,确定有限正则性在内部强制为零或满足协变微分方程,并给出Graham型定理。
We prove that the Shilov boundary is invariant under proper holomorphic mappings between some classes of domains (containing among others quasi-balanced domains with the continuous Minkowski functionals). Moreover, we obtain an extension theorem for proper holomorphic mappings between quasi-circular domains. Using these results we show that there are no non-trivial proper holomorphic self-mappings in the tetrablock. Another important result of our work is a description of Shilov boundaries of a large class of domains (containing among other the symmetrized polydisc and the tetrablock). It is a…
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