Jean-Louis Clerc, Karl-Hermann Neeb
Let ${\cal D}$ be a bounded symmetric domain of tube type, $S$ its Shilov boundary, and $G$ the neutral component of its group of biholomorphic transforms. We classify the orbits of $G$ in the set $S\times S\times S$.
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We prove that irreducible symmetric domains are uniquely determined by the homotopy equivalence classes of their Shilov boundaries.
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.
We show that there are many sets in the boundary of a bounded symmetric domain that determine the values and norm of holomorphic functions on the domain having continuous extensions to the boundary. We provide an analogue of the Bergmann-Shilov boundary for finite rank JB*-triples.
Rigidity of holomorphic maps between Shilov boundaries of type-I bounded symmetric domains
该研究在复格拉斯曼流形上引入正交结构与正交映射，推广I型有界对称域Shilov边界间全纯映射，得到刚性结果，还建立了秩1到更高秩I型域Shilov边界全纯映射的刚性结论并证明其最优性。
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。
The Bergman-Shilov boundary for subfamilies of $q$-plurisubharmonic functions
本文为紧致Hausdorff空间上某些上半连续函数子类引入Bergman-Shilov边界，证明其存在唯一性，建立与峰值点及Bishop型定理的联系，推广Bychkov定理，并证明光滑伪凸域上$q$-多次调和函数Shilov边界的部分具有$q$维复子流形叶状结构。
Gap phenomenon of holomorphic maps between Shilov boundaries of type-I bounded symmetric domains
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。
Our first main result gives assumptions guaranteeing that proper holomorphic maps between Cartan type I bounded symmetric domains have simple block matrix shape, answering positively a question of Mok. The proof is based on the second main result establishing similar phenomenon for local CR maps between arbitrary boundary components of two bounded symmetric domains of the above type. Since boundary components other than Shilov boundaries are Levi-degenerate, our analysis is based on their $2$-nondegeneracy combining Levi forms with higher order tensors.
Motivated by the Shilov boundaries of bounded symmetric domains we consider arbitrary CR-quadrics in a complex linear space (of finite dimension) that have a certain symmetry property. For these the non-affine local CR-automorphisms have a simple explicit description. A large class of examples is presented.
In this paper, we characterize $C^2$-smooth totally geodesic isometric embeddings $f\colon Ω\toΩ'$ between bounded symmetric domains $Ω$ and $Ω'$ which extend $C^1$-smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if $Ω$ is irreducible, there exist totally geodesic bounded symmetric subdomains $Ω_1$ and $Ω_2$ of $Ω'$ such that $f = (f_1, f_2)$ maps into $Ω_1\times Ω_2\subset Ω$ where $f_1$ is holomorphic and $f_2$ is anti-holomorphic totally geodesic isometric embeddings. If $\text{rank}(Ω')<2\text{rank}(Ω)$, then either $f$ o…
We prove a number of results on the automorphisms of and isomorphisms between Hardy-Toeplitz algebras $\mathcal{T}(D)$ associated to bounded symmetric domains $D$: that the stable isomorphism class of $\mathcal{T}(D)$ determines $D$ (even when it is reducible), that for reducible domains $D=D_1\times\cdots \times D_s$ the automorphisms of the Shilov boundary $\check{S}(D)$ induced by those of $\mathcal{T}(D)$ permute the Shilov boundaries $\check{S}(D_i)$, and that by contrast to arbitrary solvable algebras, automorphisms of $\mathcal{T}(D)$ that are trivial on their character spaces $\check{S…
Rationality of proper holomorphic maps between bounded symmetric domains of the first kind
本文研究第一类不可约有界对称域间满足边界正则性与希洛夫边界映射条件的逆紧全纯映射，给出其分解形式与有理性判定，并推导了特定秩范围下映射的分类结论。
Clark Measures Associated with Rational Inner Functions on Bounded Symmetric Domains
本文研究了有界对称域上与有理内函数相关的 Clark 测度，给出了其显式公式，并刻画了多圆盘上 $H^2(μ_α)=L^2(μ_α)$ 成立的条件及一般域的充要条件。
A High-Rank Gap Theorem for Type-I Bounded Symmetric Domains
本文针对 Type-I 有界对称域间的全纯映射，在秩间隙条件下证明映射含固定恒等块，并给出最优性例子及邻域条件必要性。
Geometric Orbital Linearization of Siegel Singularities in the Plane
本文通过复相切几何刻画了平面上 Siegel 奇点的解析线性化，提出了圆球上的强准则及多项式与对数边界判据，并研究了相切环面的横截动力学与刚性。
We give the presentation of exceptional bounded symmetric domains using the Albert algebra and exceptional Jordan triple systems. The first chapter is devoted to Cayley-Graves algebras, the second to exceptional Jordan triple systems. In the third chapter, we give a geometric description of the two exceptional bounded symmetric domains, their boundaries and their compactification.
In this paper we study some extremal problems for the family $S_g^0(\mathbb{B}_X)$ of normalized univalent mappings with $g$-parametric representation on the unit ball $\mathbb{B}_X$ of an $n$-dimensional JB$^*$-triple $X$ with $r\geq 2$, where $r$ is the rank of $X$ and $g$ is a convex (univalent) function on the unit disc $\mathbb{U}$, which satisfies some natural assumptions. We obtain sharp coefficient bounds for the family $S_g^0(\mathbb{B}_X)$, and examples of bounded support points for various subsets of $S_g^0(\mathbb{B}_X)$. Our results are generalizations to bounded symmetric domains…
Let K be any compact set in the complex plane that has a connected complement, let A(K) be the uniforn algebra of all continuous complex functions on K that are holomorphic on the interior of K, let bK be the topological boundary of K, let z be in K, and let M be the maximal ideal of functions in A(K) that are 0 at z. Using only facts from classical complex analytic function theory, and without using any results from the theory of analytic capacity, we prove the following: z is an element of bK iff z is a peak point for A(K) iff z belongs to the Shilov boundary for A(K) iff z belongs to the Bi…
Carathéodory distance-preserving maps between bounded symmetric domains
研究保Carathéodory距离的有界对称域映射的刚性，证明其在特定条件下为全纯或反全纯映射，并利用几何工具获得结果。
We introduce two classes of "egg type" domains, built on general bounded symmetric domains, for which we compute the Bergmann kernel in explicit form. We use the characterization of bounded symmetric domains through Jordan triple systems. The egg type domains are defined using the generic norm. A generalization of the Hua integral is computed; the result shows the existence of a special polynomial with integer or half-integer coefficients, attached to each irreducible bounded symmetric domain.
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