Gregery T. Buzzard, Franc Forstneric
We construct automorphisms of $\C^n$ which map certain discrete sequences one onto another with prescribed finite jet at each point, thus solving a general Mittag-Leffler interpolation problem for automorphisms. Under certain circumstances, this can be done while also approximating a given automorphism on a compact set.
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We approximate smooth maps defined on non-compact totally real manifolds by holomorphic automorphisms of $\mathbb C^n$.
We apply Nevanlinna theory for algebraic varieties to Danielewski surfaces and investigate their group of holomorphic automorphisms. Our main result states that the overshear group which is known to be dense in the identity component of the holomorphic automorphism group, is a free amalgamated product.
We show the existence of free dense subgroups, generated by 2 elements, in the holomorphic shear and overshear group of complex-Euklidean space and extend this result to the group of holomorphic automorphisms of Stein manifolds with Density Property, provided there exists a generalized translation. The conjugation operator associated to this generalized translation is hypercyclic on the topological space of holomorphic automorphisms.
It is proved that the group of holomorphic automorphisms of holomorphically homogeneous nondegenerate (finite Bloom-Graham type + holomorphic nondegenaracy) model surface Q is a subgroup of the group of birational automorphisms of the ambient space (Cremona group) with uniformly bounded degree. An estimate of the degree of the automorphisms in terms of the dimension of the ambient space is given (Theorem 4). It is proved that none of the conditions of the theorem can be omitted. We consider also the question about connectedness of the automorphism group of Q (Theorem 7).
Given a sequence of automorphisms of the polydisk, we show that the associated composition semigroup homomorphisms on the ball of bounded holomorphic functions on the polydisk admit a universal inner function if a certain condition on the automorphisms are satisfied.
We consider the problem of interpolating a holomorphic family of nondegenerate holomorphic jets on C^n for n > 1, parametrized by points in a Stein manifold, by a holomorphic family of automorphisms of C^n. We show that under a suitable topological condition it is possible to find a solution, thereby generalizing results of Forstneric and Varolin concerning the nonparametric jet interpolation by automorphisms.
The basin of attraction of a uniformly attracting sequence of holomorphic automorphisms that agree to a certain order in the common fixed point, is biholomorphic to $\mathbb{C}^n$. We also give sufficient estimates how large this order has to be.
We overview a number of precise notions for a holomorphic automorphism group to be big together with their implications, in particular we give an exposition of the notions of flexibility and of density property. These studies have their origin in the famous result of Andersén and Lempert from 1992 proving that the overshears generate a dense subgroup in the holomorphic automorphism group of $\C^n, n\ge 2$. There are many applications to natural geometric questions in complex geometry, several of which we mention here. Also the Linearization Problem, well known since the 1950 s and considered b…
We deal with a problem of the reconstruction of any holomorphic function $f$ on the unit ball of $\mathbb{C}^2$ from its restricions on a union of complex lines. We give an explicit formula of Lagrange interpolation's type that is constructed from the knowledge of $f$ and its derivatives on these lines. We prove that this formula approximates any function when the number of lines increases. The motivation of this problem comes also from possible applications in mathematical economics and medical imaging.
Given a Stein manifold X of dimension n>1, a discrete sequence a_j in X, and a discrete sequence b_j in C^m where m > [3n/2], there exists a proper holomorphic embedding of X into C^m which sends a_j to b_j for every j=1,2,.... This is the interpolation version of the embedding theorem due to Eliashberg, Gromov and Schurmann. The dimension m cannot be lowered in general due to an example of Forster.
Realizing semisimple Lie groups as holomorphic automorphism groups of bounded domains
本文证明了包括所有连通实半单群在内的一大类李群（含许多非线性李群）均可实现为仿射复空间有界域的全纯自同构群，推广了1990年关于连通线性李群的结论。
Approximation of biholomorphic maps between Runge domains by holomorphic automorphisms
本文研究了复仿射空间中Runge域之间biholomorphic映射的近似问题，证明其可由全纯自同构逼近，并推广了相关结果。
We provide a necessary condition for the existence of a 3-point holomorphic interpolant $F:\mathbb{D}\longrightarrowΩ_n$, $n\geq 2$. Our condition is inequivalent to the necessary conditions hitherto known for this problem. The condition generically involves a single inequality and is reminiscent of the Schwarz lemma. We combine some of the ideas and techniques used in our result on the $\mathcal{O}(\mathbb{D},Ω_n)$-interpolation problem to establish a Schwarz lemma -- which may be of independent interest -- for holomorphic correspondences from $\mathbb{D}$ to a general planar domain $Ω\Subset…
We show that the group of holomorphic automorphisms of a Stein manifold X of dimension greater than 1 is infinite-dimensional, provided X is a homogeneous space of a holomorphic action of a complex Lie group.
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.
Holomorphic Factorization of Vector Bundle Automorphisms
该研究在复几何领域证明了Stein空间上秩2向量丛的零伦特殊全纯自同构可分解为有限个幂幺自同构或指数映射的乘积，推进了全纯向量丛自同构的结构研究。
We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-2$. This result complements an existing classification for automorphism group dimension $n^2-1$ and greater obtained without the homogeneity assumption.
Let $f$ be a $r\times m$-matrix of holomorphic functions that is generically surjective. We provide explicit integral representation of holomorphic $ψ$ such that $ϕ=fψ$, provided that $ϕ$ is holomorphic and annihilates a certain residue current with support on the set where $f$ is not surjective. We also consider formulas for interpolation. As applications we obtain generalizations of various results previously known for the case $r=1$.
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.
Consider a holomorphic automorphism acting hyperbolically on an invariant compact set. It has been conjectured that the arising stable manifolds are all biholomorphic to Euclidean space. Such a stable manifold is always equivalent to the basin of a uniformly attracting sequence of maps. The equivalence of such basins to Euclideans has been shown under various additional assumptions. Recently Majer and Abbondandolo achieved new results by non-autonomously conjugating to normal forms on larger and larger time intervals. We show here that their results can be improved by adapting these time inter…
Approximation of holomorphic Legendrian curves with jet-interpolation
该文证明全纯勒让德曲线的插值与逼近定理，包括可数点集位于浸入曲面及Carleman型逼近，并给出适当嵌入条件。
Parametric Symplectic Jet Interpolation
本文证明了带有Stein空间参数的$\mathbb{C}^{2n}$辛全纯自同构的参数化喷射插值定理，并提供了辛全纯映射不可避免集的实例。
This paper reports on constructive approximation methods for three classes of holomorphic functions on the unit disk which are closely connected each other: the class of starlike and spirallike functions, the class of semigroup generators, and the class of functions with positive real part. It is more-or-less known that starlike or spirallike functions can be defined as solutions of singular differential equations which, in general, are not stable under the motion of interior singular points to the boundary. At the same time, one can establish a perturbation formula which continuously transfor…
We show that the group of all holomorphic automorphisms of complex affine space $\mathbb C^n$, $n>1$, and several of its subgroups satisfy the parametric Oka property with approximation and with interpolation on discrete sets.
Let $X$ be a Stein manifold of dimension $n\geq 2$ satisfying the volume density property with respect to an exact holomorphic volume form. For example, $X$ could be $\mathbb{C}^n$, any connected linear algebraic group that is not reductive, the Koras-Russell cubic, or a product $Y\times\mathbb{C}$, where $Y$ is any Stein manifold with the volume density property. We prove that chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of $X$. In particular, $X$ has a chaotic holomorphic automorphism. A proof for $X=\mathbb{C}^n$ may be found in work of Fornæss and Sib…
Coherent state representations of the holomorphic automorphism group of a quasi-symmetric Siegel domain of type III
本文研究了III型拟对称Siegel域全纯自同构群的泛相干态表示分类，揭示了其与Kähler代数结构理论的联系，并提出基于广义Iwasawa分解的方法。
Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$.
A complex manifold $Y$ is said to have the interpolation property if a holomorphic map to $Y$ from a subvariety $S$ of a reduced Stein space $X$ has a holomorphic extension to $X$ if it has a continuous extension. Taking $S$ to be a contractible submanifold of $X=\mathbb{C}^n$ gives an ostensibly much weaker property called the convex interpolation property. By a deep theorem of Forstnerič, the two properties are equivalent. They (and about a dozen other nontrivially equivalent properties) define the class of Oka manifolds. This paper is the first attempt to develop Oka theory for singular tar…
Answering questions posed in arXiv:1403.1960 we determine the group of holomorphic automorphisms of the pentablock and study its geometry
We construct holomorphic families of proper holomorphic embeddings of $\C^k$ into $\C^n$ ($0<k<n-1$), so that for any two different parameters in the family no holomorphic automorphism of $\C^n$ can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of $\C^n$ we derive the existence of families of holomorphic $\C^*$-actions on $\C^n$ ($n\ge 5$) so that different actions in the family are not conjugate. This result is surprising in view of the long standing Holomorphic Linearization Problem, which in partic…
We study fixed point sets for holomorphic automorphisms (and endomorphisms) on complex manifolds. The main object of our interest is to determine the number and configuration of fixed points that forces an automorphism (endomorphism) to be the identity. These questions have been examined in a number of papers for a bounded domain in ${\Bbb C}^n$. Here we resolve the case for a general finite dimensional hyperbolic manifold. We also show that the results for non-hyperbolic manifolds are notably different.
We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works of D. Guan and the first author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kähler manifold $W_F$ which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction, and prove that the automorphism group of $Q$ satisfies the Jordan proper…
We prove an analogue of Alexander's Theorem for holomorphic mappings of the unit ball in a complex Hilbert space: Every holomorphic mapping which takes a piece of the boundary of the unit ball into the boundary of the unit ball and whose differential at some point of this boundary is onto is the restriction of an automorphism of the ball. We also show that it is enough to assume that the mapping is only Gateaux-holomorphic.
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 study the approximation of maps into complex manifolds along with interpolation on certain compact subsets of the plane. Results are also obtained regarding approximation and interpolation of sections of holomorphic submersions.
We give here some precisions and improvements about the validity of the explicit reconstruction of any holomorphic function on a ball of $\mathbb{C}^2$ from its restrictions on a family of complex lines. Such validity depends on the mutual distribution of the lines. This condition can be geometrically described and is equivalent to a stronger stability of the reconstruction formula in terms of permutations and subfamilies of these lines. The motivation of this problem comes from possible applications in mathematical economics and medical imaging.
Holomorphic automorphisms of Markov-type surfaces
本文证明正规约化仿射复簇上奇异辛版Andersén–Lempert定理，并据此刻画Markov型曲面的全纯辛自同构，表明有序Markov三元组的任意单射自映射均可被实现。
We show that if the group of holomorphic automorphisms of a connected Stein manifold $M$ is isomorphic to that of ${\bf C}^n$ as a topological group equipped with the compact-open topology, then $M$ is biholomorphically equivalent to ${\bf C}^n$.
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.
We study and generalize a classical theorem of L. Bers that classifies domains up to biholomorphic equivalence in terms of the algebras of holomorphic functions on those domains. Then we develop applications of these results to the study of domains with noncompact automorphism group.
Compact, connected, complex manifolds that admit a compact transitive group of holomorphic automorphisms
本文提出一种仅基于李理论标准结论的李代数方法，为紧复齐性流形相关经典结论给出新证明，还提供了将典范阿贝尔李代数与紧李代数上可积复结构关联的方式，拓展了前人研究。
We introduce the class of $n$-extremal holomorphic maps, a class that generalises both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrised bidisc $Γ$. We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence $\mathcal{C}_ν, ν\geq 0,$ of necessary conditions for solvability, prove that they are of strictly increasing strength and show…
Rigidity of proper holomorphic self-mappings of the hexablock
本文研究六块域上的真全纯自映射，证明其必为自同构，从而解决了 Biswas-Pal-Tomar 提出的自同构群结构猜想。
There are three new things in this paper about the open symmetrized bidisk $\mathbb G = \{(z_1+z_2, z_1z_2) : |z_1|, |z_2| < 1\}$. They are motivated in the Introduction. In this Abstract, we mention them in the order in which they will be proved. \begin{enumerate} \item The Realization Theorem: A realization formula is demonstrated for every $f$ in the norm unit ball of $H^\infty(\mathbb G)$. \item The Interpolation Theorem: Nevanlinna-Pick interpolation theorem is proved for data from the symmetrized bidisk and a specific formula is obtained for the interpolating function. \item The Extensio…
Equidistribution of saddle periodic points for automorphisms on compact Kähler manifolds
本文证明了紧致Kähler流形上具有简单上同调作用的全纯自同构的鞍型周期点关于平衡测度等分布，并推广到正拓扑熵的Kähler曲面情形。
Holomorphic Interpolation of Multivariate Completely Monotone Functions
通过将完全单调函数表示为正测度的Laplace或Stieltjes-Fantappiè变换，利用非交换Radon变换框架结合矩阵束实现与Weyl运算微积，实现有限点插值，得到方向完全单调的整函数或有理函数。
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。
Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. We prove that on any closed discrete subset of $M$ one can prescribe the values of a conformal minimal immersion $M\to\mathbb{R}^n$. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the values of the generalized Gauss map. Furthermore, the interpolating immersions can be chosen to be complete, proper into $\mathbb{R}^n$ if the prescription of values is proper, and one-to-one if $n\ge 5$ and the prescription of values is one-to-one. We may also prescribe the flu…
We analyse the 3-extremal holomorphic maps from the unit disc $\mathbb{D}$ to the symmetrised bidisc $ \mathcal{G}$, defined to be the set $ \{(z+w,zw): z,w\in\mathbb{D}\}$, with a view to the complex geometry and function theory of $\mathcal{G}$. These are the maps whose restriction to any triple of distinct points in $\mathbb{D}$ yields interpolation data that are only just solvable. We find a large class of such maps; they are rational of degree at most 4. It is shown that there are two qualitatively different classes of rational $\mathcal{G}$-inner functions of degree at most 4, to be call…
In this article, we consider an infinite type domain $Ω_P$ in $\mathbb C^2$. The purpose of this paper is to investigate the holomorphic vector fields tangent to an infinite type model in $\mathbb C^2$ vanishing at an infinite type point and to give an explicit description of the automorphism group of $Ω_P$.
We classify polynomial models for real hypersurfaces in $\mathbb C^N$, which admit nonlinearizable infinitesimal CR automorphisms. As a consequence, this provides an optimal 1-jet determination result in the general case. Further we prove that such automorphisms arise from one common source, by pulling back via a holomorphic mapping a suitable symmetry of a hyperquadric in some complex space.
Given a convergent sequence of nodes we present a one-dimensional-holomorphic-function version of the Newton interpolation method of polynomials. It also generalises the Taylor and the Laurent formula. In other words, we present an effective identity theorem for complex domains.
A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that U(1) acts on an LCK manifold M by holomorphic isometries, and the lifting of this action to the Kahler cover of M is not isometric. We show that the cover admits an automorphic Kahler potential, and hence can be embedded to a Hopf manifold.
We prove that the Euclidean ball can be realized as a Fatou component of a holomorphic automorphism of $\mathbb{C}^m$, in particular as the escaping and the oscillating wandering domain. Moreover, the same is true for a large class of bounded domains, namely for all bounded regular open sets $Ω\subset \mathbb{C}^m$ whose closure is polynomially convex. Our result gives in particular the first example of a bounded Fatou component with a smooth boundary in the category of holomorphic automorphisms.
In this article, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows, generalizing a result of Ransford-White for the spectral unit ball, that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider $2$-point and $3$-point Pick-Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary cond…
It is an elementary fact that the action by holomorphic automorphisms on C^n is infinitely transitive, i.e., m-transitive for any m in N. The same holds on any Stein manifold with the holomorphic density property X. We study a parametrized case: we consider m points on X parametrized by a Stein manifold W, and seek a family of automorphisms of X, parametrized by W, putting them into a standard form which does not depend on the parameter. This general transitivity is shown to enjoy an Oka principle, to the effect that the obstruction to a holomorphic solution is of a purely topological nature. …
We give a sufficient condition for the abstract basin of attraction of a sequence of holomorphic self-maps of balls in \mathbb{C}^{d} to be biholomorphic to \mathbb{C}^{d}. As a consequence, we get a sufficient condition for the stable manifold of a point in a compact hyperbolic invariant subset of a complex manifold to be biholomorphic to a complex Euclidean space. Our result immediately implies previous theorems obtained by Jonsson-Varolin and by Peters; in particular, we prove (without using Oseledec's theory) that the stable manifold of any point where the negative Lyapunov exponents are w…
We prove closing lemmas for automorphisms of a Stein manifold with the density property and for endomorphisms of an Oka-Stein manifold. In the former case we need to impose a new tameness condition. It follows that hyperbolic periodic points are dense in the tame non-wandering set of a generic automorphism of a Stein manifold with the density property and in the non-wandering set of a generic endomorphism of an Oka-Stein manifold. These are the first results about holomorphic dynamics on Oka manifolds. We strengthen previous results of ours on the existence and genericity of chaotic volume-pre…
Answering a question of Lindholm, we prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density. The techniques combine methods from several complex variables (estimates for $\bar \partial$) and the theory of localized frames in general reproducing kernel Hilbert…
A holomorphic mapping $H$ between two real-analytic CR manifolds $M$ and $M'$ is said to be locally rigid if any other holomorphic map $F\colon M \to M'$ which is close enough to $H$ is obtained by composing $H$ with suitable automorphisms of $M$ and $M'$. With the aim of reducing the local rigidity problem to a linear one, we provide sufficient infinitesimal conditions. Furthermore we study some topological properties of the action of the automorphism group on the space of nondegenerate mappings from $M$ to $M'$.
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