A sequence of points $z_k$ in the unit disk is said to be thin for a given decrease function $ρ$, if there is a nontrivial bounded holomorphic function such that the infinite series $\sum_k ρ(1-|z_k|)|f(z_k)|$ converges. All sequences will be assumed hyperbolically separated. We give necessary and sufficient conditions for the problem of thinness of a sequence to be non-trivial (one way or the other), and for two different decrease functions to give rise to the same thin sequences. Along the way, some concrete conditions (necessary or sufficient) for a sequence to be thin are obtained.
We show that the topological equivalence class of holomorphic foliation germs with an isolated singularity of Poincaré type is determined by the topological equivalence class of the real intersection foliation of the (suitably normalized) foliation germ with a sphere centered in the singularity. We use this Reconstruction Theorem to completely classify topological equivalence classes of plane holomorphic foliation germs of Poincaré type and discuss a conjecture on the classification in dimension $\geq 3$.
The following result is proved: Let $D$ and $D'$ be bounded domains in $\mathbb C^n$, $\partial D$ is smooth, real-analytic, simply connected, and $\partial D'$ is connected, smooth, real-algebraic. Then there exists a proper holomorphic correspondence $f:D\to D'$ if and only if $\partial D$ and $\partial D'$ a locally CR-equivalent. This leads to a characterization of the equivalence relationship between bounded domains in $\mathbb C^n$ modulo proper holomorphic correspondences in terms of CR-equivalence of their boundaries.
We consider (small) algebraic deformations of germs of real-algebraic CR submanifolds in complex space and study the biholomorphic equivalence problem for such deformations. We show that two algebraic deformations of minimal holomorphically nondegenerate real-algebraic CR submanifolds are holomorphically equivalent if and only if they are algebraically equivalent.
We introduce several homotopy equivalence relations for proper holomorphic mappings between balls. We provide examples showing that the degree of a rational proper mapping between balls (in positive codimension) is not a homotopy invariant. In domain dimension at least 2, we prove that the set of homotopy classes of rational proper mappings from a ball to a higher dimensional ball is finite. By contrast, when the target dimension is at least twice the domain dimension, it is well known that there are uncountably many spherical equivalence classes. We generalize this result by proving that an a…
We show that a pseudo-holomorphic embedding of an almost-complex $2n$-manifold into almost-complex $(2n + 2)$-Euclidean space exists if and only if there is a CR regular embedding of the $2n$-manifold into complex $(n + 1)$-space. We remark that the fundamental group does not place any restriction on the existence of either kind of embedding when $n$ is at least three. We give necessary and sufficient conditions in terms of characteristic classes for a closed almost-complex 6-manifold to admit a pseudo-holomorphic embedding into $\R^8$ equipped with an almost-complex structure that need not be…
We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $2$.
We consider inductive limits of weighted spaces of holomorphic functions in the unit ball of $\mathbb C^n$. The relationship between sets of uniqueness, weakly sufficient sets and sampling sets in these spaces is studied. In particular, the equivalence of these sets, under general conditions of the weights, is obtained.
We give several characterizations of holomorphic mean Besov-Lipschitz space on the unit ball in $\cn $ and appropriate Besov-Lipschitz space and prove the equivalences between them. Equivalent norms on the mean Besov-Lipschitz space involve different types of $L^p$-moduli of continuity, while in characterizations of Besov-Lipschitz space we use not only the radial derivative but also the gradient and the tangential derivatives. The characterization in terms of the best approximation by polynomials is also given.
We construct Coleff-Herrera products and Bochner-Martinelli type residue currents associated with a tuple $f$ of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case, as the transformation law, the Poincaré-Lelong formula and the equivalence of the Coleff-Herrera product and the Bochner-Martinelli type residue current associated with $f$ when $f$ defines a complete intersection.
We consider an embedded $n$-dimensional compact complex manifold in $n+d$ dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert's formal principle program. We will give conditions ensuring that a neighborhood of $C_n$ in $M_{n+d}$ is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold's result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in $M_{n+d }$ having $C_n$ as a compact leaf, extending Ueda's theory to the high codimens…
We give a proof of the following theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., 2 (1989), 851-897). Let Z be a holomorphic fiber bundle over a Stein manifold. If the fiber of Z admits a dominating holomorphic spray then the inclusion of the space of global holomorphic sections of Z into the space of continuous sections is a weak homotopy equivalence. In the classical case when the fiber is a complex Lie group or a homogeneous space this was proved by H. Grauert (Math. Ann., 135 (1958), 263--273). For further results in this direction se…
For a harmonic diffeomorphism between the Poincaré disks, Wan showed the equivalence between the boundedness of the Hopf differential and the quasi-conformality. In this paper, we will generalize this result from quadratic differentials to $r$-differetials. We study the relationship between bounded holomorphic $r$-differentials and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space $SL(r,\mathbb R)/SO(r)$ arising from cyclic/subcyclic harmonic Higgs bundles. Also, we show the equivalences between the boundedness of holomorphic differentials and havi…
There are solved standard problems related to Formal (Holomorphic) Segre preserving Mappings of non-trivial Real-Formal Hypersurfaces in $\mathbb{C}^{2}$.
The purpose of this paper is to investigate the geometric properties of real hypersurfaces of D'Angelo infinite type in ${\mathbb C}^n$. In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using Newton polyhedra,which is an important concept in singularity theory. More precisely,equivalence conditions are given in the case of some model hypersurfaces.
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…
Let $(W,Π)$ be a Riemann domain over a complex manifold $M$ and $w_0$ be a point in $W$. Let $\mathbb D$ be the unit disk in $\mathbb C$ and $\mathbb T=\bd\mathbb D$. Consider the space ${\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ of continuous mappings $f$ of $\mathbb T$ into $W$ such that $f(1)=w_0$ and $Π\circ f$ extends to a holomorphic on $\mathbb D$ mapping $\hat f$. Mappings $f_0,f_1\in{\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ are called {\it $h$-homotopic} if there is a continuous mapping $f_t$ of $[0,1]$ into $\rS_{1,w_0}({\bar{\mathbb D}},W,M)$. Clearly, the $h$-homotopy is an eq…
We study on the biholomorphic equivalence of a strongly pseudoconvex bounded domain with differentiable spherical boundary to an open ball, and we study on the biholomorphicity of a proper holomorphic self mapping of a strongly pseudoconvex domain with real analytic boundary.
We construct an injective map from the set of holomorphic equivalence classes of neighborhoods $M$ of a compact complex manifold $C$ into ${\mathbb C}^m$ for some $m<\infty$ when $(TM)|_C$ is fixed and the normal bundle of $C$ in $M$ is either weakly negative or $2$-positive.
For any open, connected and bounded set $Ω\subseteq \mathbb C^m$, let $\mathcal A$ be a natural function algebra consisting of functions holomorphic on $Ω$. Let $\mathcal M$ be a Hilbert module over the algebra $\mathcal A$ and $\mathcal M_0\subseteq \mathcal M$ be the submodule of functions vanishing to order $k$ on a hypersurface $\mathcal Z \subseteq Ω$. Recently the authors have obtained an explicit complete set of unitary invariants for the quotient module $\mathcal Q = \mathcal M \ominus \mathcal M_0$ in the case of $k=2$. In this paper, we relate these invariants to familiar notions fro…
On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n>1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local CR invariant of the boundary. For a bounded strictly pseudoconvex domain $Ω\subset \mathbb{C}^2$ diffeomorphic to the ball, we prove that the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball, subject to the existence of a holomorphic vector field satisfying a mild approximate tangency condition along the boundary. In particular, by considering the Euler vector …
In this article we consider the following equivalence relation on the class of all functions of two variables on a set $X$: we will say that $L,M: X\times X\to \mathbb{C}$ are rescalings if there are non-vanishing functions $f,g$ on $X$ such that $M\left(x,y\right)=f\left(x\right)g\left(y\right) L\left(x,y\right)$, for any $x,y\in X$. We give criteria for being rescalings when $X$ is a topological space, and $L$ and $M$ are separately continuous, or when $X$ is a domain in $\mathbb{C}^{n}$ and $L$ and $M$ are sesqui-holomorphic. A special case of interest is when $L$ and $M$ are symmetric, and…
Characterizations of Hardy spaces on tube domains over polyhedral cones
Department of Mathematics, Linyi University（临沂大学）；Department of Mathematics, University of Missouri（密苏里大学）；Department of Mathematics, Zhejiang University（浙江大学）
Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their "dual" objects, the plurisubharmonic functions. These functions are defined classically by requiring that the restriction to each pseudo-holomorphic curve is subharmonic. In this paper subharmonic functions are defined by applying the viscosity approach to a version of the complex hessian which exists intrinsically on any almost complex manifold. Three theorems are proved. The first is a restriction theorem which establishes the equivalence of our definition with the "classical" definition. I…
