On automorphisms of the symmetrized bidisc

Bergman metric on the symmetrized bidisc and its consequences

On the symmetrized bidisc G2 with the Bergman metric, the holomorphic sectional curvature is negatively pinched and the holomorphic bisectional curvature is not. The consequences in invariant metrics are provided.

对称双圆盘上的等距映射，II

Isometries in the symmetrized bidisc, II

该研究证明了保持Kobayashi距离的从单位圆盘D到对称双圆盘G₂的映射f必为全纯或反全纯，完善了对称双圆盘上等距映射的相关理论。

Nevanlinna-Pick problem and uniqueness of left inverses in convex domains, symmetrized bidisc and tetrablock

In the paper we discuss the problem of uniqueness of left inverses (solutions of two point Nevanlinna-Pick problem) in bounded convex domains, strongly linearly convex domains, the symmetrized bidisc and the tetrablock.

Spectral Nevanlinna-Pick problem and weak extremals in the symmetrized bidisc

The main goal of the paper is to study $m$-extremal mappings in the symmetrized bidisc showing that they are rational and $\GG_2$-inner which, in particular, answers a question posed in \cite{Agl-Lyk-You 2013}.

对称化斜双曲盘上全纯函数的模型

Models of holomorphic functions on the symmetrized skew bidisc

本文研究对称化斜双曲盘上模不超过1的全纯函数理论，提出函数的实现公式和模型公式。

Nonuniqueness of Carathéodory extremal functions on the symmetrized bidisc

We survey the Carathéodory extremal problem $\mathrm{Car} δ$ on the symmetrized bidisc $$ G = \{(z+w,zw):|z|<1, \, |w|<1\} = \{(s,p)\in \mathbb{C}^2: |s-\bar s p| < 1-|p|^2\}. $$ We also give some new results on this topic. We are particularly interested in cases of this problem in which the solution of the problem is not unique. It is known that, for any $δ=(λ,v)\in TG$ with $v\neq 0$, there is at least one $ω\in\mathbb{T}$ such that $Φ_ω$ solves $\mathrm{Car} δ$, where $Φ_ω(s,p) = \frac{2ωp-s}{2-ωs}$. Moreover, there is an essentially unique solution of $\mathrm{Car} δ$ if and only if $δ$ ha…

Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc

A set $V$ in a domain $U$ in $\mathbb{C}^n$ has the {\em norm-preserving extension property} if every bounded holomorphic function on $V$ has a holomorphic extension to $U$ with the same supremum norm. We prove that an algebraic subset of the {\em symmetrized bidisc} \[ G := \{(z+w,zw):|z|<1, |w| < 1 \} \] has the norm-preserving extension property if and only if it is either a singleton, $G$ itself, a complex geodesic of $G$, or the union of the set $\{(2z,z^2): |z|<1\}$ and a complex geodesic of degree $1$ in $G$. We also prove that the complex geodesics in $G$ coincide with the nontrivial h…

关于对称双圆盘的几何

On the geometry of the symmetrized bidisc

本文研究对称双圆盘自同构群作用及其几何，证明它双全纯于 Isaev 分类中的 $\mathcal D_1$，并给出二者新刻画及相关域到双圆盘乘积的显式双全纯映射。

关于某些非凸域的弱沃尔夫-登joy定理

On weak Wolff--Denjoy theorem for certain non-convex domains

本文研究了某些非凸域上holomorphic自映射的固定点或紧致发散序列性质，揭示了对称双复盘、对称三复盘、四块和五块等属于此类域，并描述了这些域上holomorphic自映射的固定点集及目标集。

通过双圆盘自同构群的一个子群刻画双圆盘

A characterization of the bidisc by a subgroup of its automorphism group

本文利用双圆盘自同构群中同构于单位圆盘自同构群的三维不可解子群及其轨道结构，给出双圆盘的双全纯刻画，并描述相关子域的自同构群。

Intrinsic Directions, Orthogonality and Distinguished Geodesics in the Symmetrized Bidisc

The symmetrized bidisc \[ G \stackrel{\rm{def}}{=}\{(z+w,zw):|z|<1,\ |w|<1\}, \] under the Carathéodory metric, is a complex Finsler space of cohomogeneity $1$ in which the geodesics, both real and complex, enjoy a rich geometry. As a Finsler manifold, $G$ does not admit a natural notion of angle, but we nevertheless show that there {\em is} a notion of orthogonality. The complex tangent bundle $TG$ splits naturally into the direct sum of two line bundles, which we call the {\em sharp} and {\em flat} bundles, and which are geometrically defined and therefore covariant under automorphisms of $G…

一族 Lempert 域

A family of Lempert domains

本文证明 Ghosh 与 Zwonek 引入的域族 $\bL_n$（第四类 Cartan 域的 2-proper 全纯像）全部是 Lempert 域，推广了对称化双盘与四面块的已知结果。

Characterizations of some domains via Carathéodory extremals

In this paper we characterize the unit disc, the bidisc and the symmetrized bidisc \[ G =\{(z+w,zw):|z|<1,\ |w|<1\} \] in terms of the possession of small classes of analytic maps into the unit disc that suffice to solve all Carathéodory extremal problems in the domain.

一致代数与特殊簇

Uniform algebras and distinguished varieties

本文揭示了特殊簇与双圆盘上一致代数的联系，证明了真多项式映射像域上Samuelsson-Wold结果的类似结论，并描述了相应多项式凸包。

与五块相关的算子及其与双球和对称双盘的关系

Operators associated with the pentablock and their relations with biball and symmetrized bidisc

本文研究了pentablock上算子的性质，探讨了P-contraction、P-unitary和P-isometry的特征，以及它们在不同领域间的相互作用。

On a domain in C^2 with generic piecewise smooth Levi-flat boundary and non-compact automorphism group

In this paper, we prove that if D is a simply-connected domain in C^2 with generic piecewise smooth Levi-flat boundary and non-compact automorphism group, then D is biholomorphic to the bidisc. The proof is based on a careful analysis of invariant measures.

An example of a bounded $\Bbb C$-convex domain which is not biholomoprhic to a convex domain

We show that the symmetrized bidisc is a $\Bbb C$-convex domain. This provides an example of a bounded $\Bbb C$-convex domain which cannot be exhausted by domains biholomorphic to convex domains.

Proper holomorphic mappings between symmetrized ellipsoids

We characterize the existence of proper holomorphic mappings in the special class of bounded $(1,2,...,n)$-balanced domains in $\mathbb{C}^n$, called the symmetrized ellipsoids. Using this result we conclude that there are no non-trivial proper holomorphic self-mappings in the class of symmetrized ellipsoids. We also describe the automorphism groupof these domains.

经典Cartan域的$2$-固有全纯像

$2$-proper holomorphic images of classical Cartan domains

本文提出研究有界对称域$2$-固有全纯像的一般方法，定义了新区域族$\mathbb L_n$，统一刻画其函数几何性质，证明其非凸穷尽性、非齐性及非Lu Qi-Keng域特征。

A Schwarz lemma for the symmetrized polydisc via estimates on another family of domains

We make some sharp estimates to obtain a Schwarz lemma for the \textit{symmetrized polydisc} $\mathbb G_n$, a family of domains naturally associated with the spectral interpolation, defined by \[ \mathbb G_n :=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j \dots, \prod_{i=1}^n z_i \right): \,|z_i|<1, i=1,\dots,n \right \}. \] We first make a few estimates for the \textit{the extended symmetrized polydisc} $\widetilde{\mathbb G}_n$, a family of domains introduced in \cite{pal-roy 4} and defined in the following way: \begin{align*} \widetilde{\mathbb G}_n := \Bigg\{ (y_1,\do…

Rigidity of Proper Holomorphic Self-mappings of the Pentablock

The pentablock is a Hartogs domain over the symmetrized bidisc. The domain is a bounded inhomogeneous pseudoconvex domain, and does not have a $\mathcal{C}^{1}$ boundary. Recently, Agler-Lykova-Young constructed a special subgroup of the group of holomorphic automorphisms of the pentablock, and Kosiński completely described the group of holomorphic automorphisms of the pentablock. The purpose of this paper is to prove that any proper holomorphic self-mapping of the pentablock must be an automorphism.

Operator theory and representation of distinguished varieties in the symmetrized tridisc

We show that every distinguished variety in the symmetrized tridisc $\mathbb G_3$ is one-dimensional and can be represented as \begin{equation}\label{eqn:1} Λ=\{ (s_1,s_2,p)\in \mathbb G_3 \,:\, (s_1,s_2) \in σ_T(F_1^*+pF_2\,,\, F_2^*+pF_1) \}, \end{equation} where $F_1,F_2$ are commuting square matrices of the same order satisfying $[F_1^*,F_1]=[F_2^*,F_2]$ and a norm condition. The converse also holds, i.e, a set of the form (\ref{eqn:1}) is always a distinguished variety in $\mathbb G_3$. We show that for a triple of commuting operators $Σ= (S_1,S_2,P)$ having $Γ_3$ as a spectral set, there…

The fibred density property and the automorphism group of the spectral ball

We generalize the notion of the density property for complex manifolds to holomorphic fibrations, and introduce the notion of the fibred density property. We prove that the natural fibration of the spectral ball over the symmetrized polydisc enjoys the fibred density property and describe the automorphism group of the spectral ball.

五角块的 Toeplitz corona 定理及其应用

A Toeplitz corona theorem for the pentablock and applications

本文陈述并证明了五角块上的 Toeplitz corona 定理，并通过两个应用得到了双圆盘和对称双圆盘 Toeplitz corona 定理的新刻画。

环面压缩与$\boldsymbol{\boldsymbol{\boldsymbol{Γ}}}$-可分辨$\boldsymbol{\boldsymbol{\boldsymbol{Γ}}}$-压缩

The toral contractions and $Γ$-distinguished $Γ$-contractions

本文研究环面压缩与$Γ$-可分辨$Γ$-压缩算子，给出两类算子可膨胀为对应等距算子的充要条件，确定双圆盘与对称化双圆盘中可分辨簇的可分辨边界，并通过示例对比两类算子理论的差异。