A. V. Isaev, N. G. Kruzhilin
We characterize pairs of bounded Reinhardt domains in $\CC^2$ between which there exists a proper holomorphic map and find all proper maps that are not elementary algebraic.
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We describe all possibilities of existence of non-elementary proper holomorphic maps between non-hyperbolic Reinhardt domains in $\mathbb C^2$ and the corresponding pairs of domains.
In this paper we characterize compact Hankel operators with conjugate holomorphic symbols on the Bergman space of bounded convex Reinhardt domains in $\mathbb{C}^2$. We also characterize compactness of Hankel operators with conjugate holomorphic symbols on smooth bounded pseudoconvex complete Reinhardt domains in $\mathbb{C}^2$.
A complete characterization of proper holomorphic mappings between domains from the class of all pseudoconvex Reinhardt domains in $\C^2$ with the logarithmic image equal to a strip or a half-plane is given.
On complete pseudoconvex Reinhardt domains in $\mathbb{C}^2$, we show that there is no nonzero Hankel operator with an anti-holomorphic symbol that is Hilbert-Schmidt. We also present examples of unbounded non-pseudoconvex domains that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols.
On horn-shaped Reinhardt domains
本文在n≥2的复空间ℂⁿ中引入一类新的无界Reinhardt域（角形域），研究其Bergman空间维数、Bergman度量全纯截面曲率及全纯等价问题。
Reinhardt domains determined by their endomorphisms
本文研究二维伪凸雷因哈特域，证明其内射映射同构时在双全息或反双全息意义上等价，并展示所有能收缩到适当嵌入空心复线的斯特恩流形由其内射映射半群确定。
We study the possible dimensions that the groups of holomorphic automorphisms of hyperbolic Reinhardt domains can have. We are particularly interested in the problem of characterizing Reinhardt domains with automorphism group of prescribed dimension.
We present an elementary proof of the cross theorem in the case of Reinhardt domains. The results illustrates the well-known interrelations between the holomorphic geometry of a Reinhardt domain and the convex geometry of its logarithmic image.
Let $Ω$ be an arbitrary bounded semi-Reinhardt domain in $\mathbb{C}^{m+n}$. We show that for $m \geq 2$, if a Hankel operator with an anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space $L_a^2(Ω)$, then it must equal zero. This fact has previously been proved for Reinhardt domains.
Unbounded Reinhardt domains with finite-dimensional Bergman spaces in $\C^n$
本文研究了具有有限维伯格曼空间的无界雷纳尔德域，证明其伯格曼度量具有正曲率且自同构群为线性映射。
It is shown that the Laurent series of a holomorphic function smooth up to the boundary on a Reinhardt domain in $\mathbb{C}^n$ converges unconditionally to the function in the Fréchet topology of the space of functions smooth up to the boundary.
On Stable Univalence and Coefficient Estimates for a Class of Pluriharmonic Mappings in Convex Reinhardt Domains
本文研究了凸Reinhardt域中双曲映射的稳定性与系数估计，通过多维Noshiro-Warschawski定理的推广，建立了双曲映射与全纯函数的一一对应关系，并给出了尖锐的系数估计。
Answering all questions---concerning proper holomorphic mappings between generalized Hartogs triangles---posed by Jarnicki and Plfug (First steps in several complex variables: Reinhardt domains, 2008) we characterize the existence of proper holomorphic mappings between generalized Hartogs triangles and give their explicit form. In particular, we completely describe the group of holomorphic automorphisms of such domains and establish rigidity of proper holomorphic self-mappings on them.
We consider a family E_m(D,M) of holomorphic bundles constructed as follows: to any given M in GL_n(Z), we associate a "multiplicative automorphism" f of (C*)^n. Now let D be a f-invariant Stein Reinhardt domain in (C*)^n. Then E_m(D,M) is defined as the flat bundle over the annulus of modulus m>0, with fiber D, and monodromy f. We show that the function theory on E_m(D,M) depends nontrivially on the parameters m, M and D. Our main result is that E_m(D,M) is Stein if and only if m log(r(M)) <= 2 π^2, where r(M) denotes the max of the spectral radii of M and its inverse. As corollaries, we: -- …
We present a result on existence of some kind of peak functions for $\C$-convex domains and for the symmetrized polydisc. Then we apply the latter result to show the equivariance of the set of peak points for $A(D)$ under proper holomorphic mappings. Additionally, we present a description of the set of peak points in the class of bounded pseudoconvex Reinhardt domains.
Let D be a piecewise smooth bounded convex Reinhardt domain in C^2. Assume that the symbols f and g are continuous on the closure of D and harmonic on the disks in the boundary of D. We show that if the product of Hankel operators H^*_f H_g is compact on the Bergman space of D, then on any disk in the boundary of D, either f or g is holomorphic.
It is shown that if the boundary of a Reinhardt domain in $\mathbb{C}^n$ contains the origin, each holomorphic function on the domain which is infinitely many times differentiable up to the boundary extends holomorphically to a neighborhood of the origin.
Recently Herbig, McNeal, and Straube have showed that the Bergman projection of conjugate holomorphic functions is smooth up to the boundary on a class of pseudoconvex domains. We show that a further smoothing property holds on a family of Reinhardt domains; namely, the Bergman projection of conjugate holomorphic functions is holomorphic past the boundary.
Local Banach Space Theoretic Approach to Bohr's Theorem for Vector Valued Holomorphic and Pluriharmonic Functions
本文通过局部巴拿赫空间理论研究向量值全纯和多值调和函数的波尔定理，证明了波尔半径的正性及渐进行为，并建立了相关系数型施瓦茨-皮克引理。
The Bohr radius for power series of holomorphic functions mapping a multidimensional Reinhardt domain into the convex domain in the complex plane is independent of this convex domain.
We show that on smooth complete Reinhardt domains, weighted Bergman projection operators corresponding to exponentially decaying weights are unbounded on $L^p$ spaces for all $p\not=2$. On the other hand, we also show that the exponentially weighted projection operators are bounded on Sobolev spaces on the unit ball.
Biholomorphic rigidity and symmetry breaking for domains with Bergman metric of constant positive holomorphic sectional curvature
本文针对Hayashida-Kamimoto构造的具有常正全纯截面曲率2的Bergman度量的Reinhardt区域，证明三明治原理并构造大量不等价光滑非Reinhardt区域，还分类双全纯映射并确定角扰动族的自同构群。
We obtain $L^p$ regularity for the Bergman projection on some Reinhardt domains. We start with a bounded initial domain $Ω$ with some symmetry properties and generate successor domains in higher {dimensions}. We prove: If the Bergman kernel on $Ω$ satisfies appropriate estimates, then the Bergman projection on the successor is $L^p$ bounded. For example, the Bergman projection on successors of strictly pseudoconvex initial domains is bounded on $L^p$ for $1<p<\infty$. The successor domains need not have smooth boundary nor be strictly pseudoconvex.
Rotational symmetries of domains and orthogonality relations
本文基于 Bergman 空间中单项式的 $L^2$ 正交关系，给出域具有 Reinhardt、圆形或 Hartogs 对称性及在环面线性群作用下不变的充分条件。
Stability of the Monomial Basis Kernel of Reinhardt domains
研究Reinhardt域中p-单项基核(p-MBK)的稳定性，证明其在参数p下连续，并探讨索引集S_p(Ω)的结构特性。
We show the following symmetry property of a bounded Reinhardt domain $Ω$ in $\mathbb{C}^{n+1}$: let $M=\partialΩ$ be the smooth boundary of $Ω$ and let $h$ be the Second Fundamental Form of $M$; if the coefficient $h(T,T)$ related to the characteristic direction $T$ is constant then $M$ is a sphere. In Appendix we state the result from an hamiltonian point of view.
Let $Ω$ be a bounded convex Reinhardt domain in $\mathbb{C}^2$ and $ϕ\in C(\barΩ)$. We show that the Hankel operator $H_ϕ$ is compact if and only if $ϕ$ is holomorphic along every non-trivial analytic disc in the boundary of $Ω$.
Abundance of Bergman metrics with constant positive holomorphic sectional curvature
本文研究了是否存在复流形其Bergman度量局部等距于m倍的Fubini-Study度量的问题，构造了Reinhardt域家族并证明了其Bergman度量的等距性，同时指出此类复流形的几何分类不可行。
Extensions of Daubechies' theorem: Reinhardt domains, Hagedorn wavepackets and mixed-state localization operators
本文扩展了Daubechies定理，研究Reinhardt领域、Hagedorn波包和混合态定位算子在量子调和分析中的应用。
We generalize Littlewood subordination principle for proper holomorphic functions and give many applications.
We prove a generalisation of Rudin's theorem on proper holomorphic maps from the unit ball to the case of proper holomorphic maps from pseudoellipsoids.
We show that there is no complete proper holomorphic map from the open disc U in C to the bidisc UxU which extends continuously to the closed disc.
For any closed analytic set X in C^2 there exists a proper holomorphic embedding of the unit disk into C^2 such that the image avoids X.
We prove that two proper holomorphic polynomial maps between bounded symmetric domains of classical type which preserve the origin are equivalent if and only if they are isotropically equivalent.
We prove that a Stein manifold of dimension $d$ admits a proper holomorphic embedding into any Stein manifold of dimension at least $2d+1$ satisfying the holomorphic density property. This generalizes classical theorems of Remmert, Bishop and Narasimhan pertaining to embeddings into complex Euclidean spaces, as well as several other recent results.
We show that each proper holomorphic self map of a symmetric power of the unit ball is an automorphism naturally induced by an automorphism of the unit ball, provided the ball is of dimension at least two.
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.
The explicit form of proper holomorphic mappings between complex ellipsoids is given. Using this description, we characterize the existence of proper holomorphic mappings between generalized Hartogs triangles and give their explicit form. In particular, the automorphism group of such domains is found.
It is shown that if a proper holomorphic map $f: \mathbb C^n \to \mathbb C^N$, $1<n\le N$, sends a pseudoconvex real analytic hypersurface of finite type into another such hypersurface, then any $n-1$ dimensional component of the critical locus of $f$ intersects both sides of $M$. We apply this result to the problem of boundary regularity of proper holomorphic mappings between bounded domains in $\mathbb C^n$.
In this paper, we prove a type of partial rigidity result for proper holomorphic mappings of certain $\ell$-concave domains in projective space into model quadratic $\ell$-concave domains. The main technical result is a degree estimate for proper holomorphic mappings into the model domains, provided that the mappings extend to projective space as rational mappings, and the source domain contains algebraic varieties and has a boundary with low CR complexity.
In 1927, Carleman showed that a continuous, complex-valued function on the real line can be approximated in the Whitney topology by an entire function restricted to the real line. In this paper, we prove a similar result for proper holomorphic embeddings. Namely, we show that a proper $\cC^r$ embedding of the real line into $\C^n$ can be approximated in the strong $\cC^r$ topology by a proper holomorphic embedding of $\C$ into $\C^n$.
The aim of this paper is to present a simple way to generate proper monomial rational maps between generalized balls and via the relations between generalized balls and bounded symmetric domains of type I, we suggest new examples of proper holomorphic maps between bounded symmetric domains of type I.
We classify proper holomorphic mappings between generalized pseudoellipsoids of different dimensions. Those domains are parametrized by the exponents. The relations among them are also obtained. Main tool is the orthogonal decomposition of a CR bundle. Such a decomposition derives the `variable-splitting' of the mapping.
We show that the only proper-holomorphic self-maps of bounded domains in C^k whose dynamics escape to a strictly pseudoconvex point of the boundary are automorphisms of the euclidean ball. This is a Wong-Rosay type result for a sequence of maps whose degrees are a priori unbounded.
A rigidity result for proper holomorphic maps between balls
本文研究了单位球之间具有大量对称性且边界上$\mathcal{C}^2$-光滑延拓的全纯真映射，证明了其刚性结果。
In this paper, we give an explicit criterion when a rational holomorphic map between balls is equivalent to a polynomial holomorphic map. Making use of this criterion, we show that any proper rational holomorphic map from B^2 into B^N of degree two is equivalent to a polynomial holomorphic map; we also construct rational holomorphic maps of degree 3 that are almost linear but are not equivalent to polynomial holomorphic maps.
We prove estimates in Hölder spaces for some Cauchy-type integral operators representing holomorphic functions in Cartesian and symmetric products of planar domains. As a consequence, we obtain information on the boundary regularity in Hölder spaces of proper holomorphic maps between symmetric products of planar domains.
We consider a proper holomorphic map form D to G domains in C^n and show that it induces a unitary isomorphism between the Bergman space A^2(G) and some subspace of A^2(D). Using this isomorphism we construct orthogonal projection onto that subspace and we derive Bells transformation formula for the Bergman kernel under proper holomorphic mappings. As a consequence of the formula we get that the tetrablock is not a Lu Qi-Keng domain.
Let $X$ be an open Riemann surface. We prove an Oka property on the approximation and interpolation of continuous maps $X \to (\mathbb{C}^*)^2$ by proper holomorphic embeddings, provided that we permit a smooth deformation of the complex structure on $X$ outside a certain set. This generalises and strengthens a recent result of Alarcon and Lopez. We also give a Forstneric-Wold theorem for proper holomorphic embeddings (with respect to the given complex structure) of certain open Riemann surfaces into $(\mathbb{C}^*)^2$.
Let f be a proper holomorphic mapping between bounded domains D and D' in C^2. Let M, M' be open pieces on the boundaries of D and D' respectively, that are smooth, real analytic and of finite type. Suppose that the cluster set of M under f is contained in M'. It is shown that f extends holomorphically across M. This can be viewed as a local version of the Diederich-Pinchuk extension result for proper mappings in C^2.
We make several new contributions to the study of proper holomorphic mappings between balls. Our results include a degree estimate for rational proper maps, a new gap phenomenon for convex families of arbitrary proper maps, and an interesting result about inverse images.
We show that any finitely connected domain $U\subset\CC$ can be properly embedded into $\CC^2$. For some sequences $\{p_j\}\subset U$, $U\setminus\{p_j\}$ can also be properly embedded into $\CC^2$.
We prove the existence of global Bishop discs in a strictly pseudoconvex Stein domain in an almost complex manifold of complex dimension 2.
In this paper we studied a broader type of generalized balls which are domains on the complex projective with possibly Levi-degenerate boundaries. We proved rigidity theorems for proper holomorphic mappings among them by exploring the structure of the moduli spaces of projective linear subspaces, which generalized some earlier results for the ordinary generalized balls with Levi-nondegenerate boundaries.
We consider proper holomorphic mappings of equidimensional pseudoconvex domains in complex Euclidean space, where both source and target can be represented as Cartesian products of smoothly bounded domains. It is shown that such mappings extend smoothly up to the closures of the domains, provided each factor of the source satisfies Condition R. It also shown that the number of smoothly bounded factors in the source and target must be the same, and the proper holomorphic map splits as product of proper mappings between the factor domains.
We show that the $\bar{\partial}$-problem is globally regular on a domain in $\mathbb{C}^n$, which is the $n$-fold symmetric product of a smoothly bounded planar domain. Remmert-Stein type theorems are proved for proper holomorphic maps between equidimensional symmetric products and proper holomorphic maps from cartesian products to symmetric products. It is shown that proper holomorphic maps between equidimensional symmetric products of smooth planar domains are smooth up to the boundary
Holomorphic retracts in the Lie ball and the tetrablock
本文研究Lempert域中全纯收缩的性质，将其存在性与线性收缩关联，并给出三维Lie球与四块体中全纯收缩的完整描述。
We prove a Fatou type theorem for bounded functions with d_J -bar differential of a controled growth on smoothly bounded domains in an almost complex manifold.
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.
We study the complexity of holomorphic isometries and proper maps from the complex unit ball to type IV classical domains. We investigate on degree estimates of holomorphic isometries and holomorphic maps with minimum target dimension. We also construct a real-parameter family of mutually inequivalent holomorphic isometries from the unit ball to type IV domains. We also provide examples of non-isometric proper holomorphic maps from the complex unit ball to classical domains.
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