Holomorphic Dependence of the Heins Map

Let D be a bounded convex domain and Hol _c (D,D) the set of holomorphic maps from D to C ⁿ with image relatively compact in D. Consider Hol _c (D,D) as a open set in the complex Banach space H ^∞ _n (D) of bounded holomorphic maps from D to C ⁿ . We show that the map τ: Hol _c (D,D) → D (called the Heins map for D equals to the unit disc of C) which associates to ? ∈ Hol _c (D,D) its unique fixed point τ? ∈ D is holomorphic and its differential is given by dτ_?(v) = (Id-df_τ(?))^?1 v(τ(?)) for v ∈ H ^∞ _n (D).     

Holomorphic curves and integral points off divisors

We deal with the distributions of holomorphic curves and integral points off divisors. We will simultaneously prove an optimal dimension estimate from above of a subvariety W off a divisor D which contains a Zariski dense entire holomorphic curve, or a Zariski dense D-integral point set, provided that in the latter case everything is defined over a number field. Then, if the number of components of D is large, the estimate leads to the constancy of such a holomorphic curve or the finiteness of such an integral point set. At the beginning, we extend logarithmic Bloch-Ochiai's Theorem to the K?hler case. Received: 10 January 2000 / Published online: 18 January 2002     

Factorisations of the Helmholtz Operator, Radó’s Theorem, and Clifford Analysis

Radó’s theorem for holomorphic functions asserts that if a continuous function is holomorphic on the complement of its zero locus, then it is holomorphic everywhere. We prove in this paper an equivalent theorem for functions lying in the kernel of a first order differential operator D{\mathcal{D}} such that the Helmholtz operator ∇²+λ can be factorized as the composition [^(D)]D{\widehat{\mathcal{D}}\mathcal{D}} . We also analyse the factorisations [^(D)]D{\widehat{\mathcal{D}}\mathcal{D}} of the Laplace and Helmholtz operators associated to the Clifford analysis and the representations of holomorphic function of several complex variables.     

Embedding Certain Infinitely Connected Subsets of Bordered Riemann Surfaces Properly into ℂ2

We prove that certain infinitely connected domains D in a bordered Riemann surface, which admits a holomorphic embedding into C ², admit a proper holomorphic embedding into C ². We also prove that certain infinitely connected open subsets D⊂ℂ admit a proper holomorphic embedding into ℂ².      

A classification of homogeneous operators in the Cowen–Douglas class

An explicit construction of all the homogeneous holomorphic Hermitian vector bundles over the unit disc D is given. It is shown that every such vector bundle is a direct sum of irreducible ones. Among these irreducible homogeneous holomorphic Hermitian vector bundles over D, the ones corresponding to operators in the Cowen–Douglas class Bn(D) are identified. The classification of homogeneous operators in Bn(D) is completed using an explicit realization of these operators. We also show how the homogeneous operators in Bn(D) split into similarity classes.     

Minimal dimension families of complex lines sufficient for holomorphic extension of functions

We consider continuous functions given on the boundary of a bounded domain D in ℂ ⁿ , n > 1, with the one-dimensional holomorphic extension property along families of complex lines. We study the existence of holomorphic extensions of these functions to D depending on the dimension and location of the families of complex lines.     

Discs in Stein manifolds containing given discrete sets

Denote by the open unit disc in . We prove that given a discrete subset S of a connected Stein manifold M there is a proper holomorphic map such that ; if the map f can be chosen to be an embedding. In addition we prove that we can prescribe higher order contacts of with given one dimensional submanifolds in M. Received: 19 June 2000; in final form: 29 November 2000 / Published online: 19 October 2001     

Local boundary regularity of holomorphic mappings

Let f be a holomorphic mapping between two bounded domains D and D' in complex space ?ⁿ. Suppose that D and D' contain smooth real hypersurfaces Γ and Γ′ as open subsets of their respective boundaries, which correspond under a continuous extension of f. We shall show that this extension is smooth, given certain restrictions on Γ, Γ, and f.     

A Note on a Theorem of H. Cartan

We prove that if D C is a bounded domain with real analytic boundary, and D is either pseudoconvex or it satisfies condition (R), then the compact open topology in Aut(D), the group of holomorphic automorphisms of D is the topology of uniform convergence on D.     

Division by a holomorphic matrix in strictly pseudoconvex Domains

LetF andG be respectively a vector- and a matrix-function in a bounded strictly pseudoconvex domainD, with entries holomorphic inD and continuous in . We prove that ifF can be divided locally byG with holomorphic factors in a neighborhood of a given pointw inD, and the rank ofG is maximal at all points of , then the division ofF byG holds globally, with some factors which are holomorphic inD and continuous in . This method applies also to other function algebras in pseudoconvex domains.     

Fredholm composition operators on spaces of holomorphic functions

Composition operators on vector spaces of holomorphic functions are considered. Necessary conditions that range of the operator is of a finite codimension are given. As a corollary of the result it is shown that a composition operatorC on a certain Banach space of holomorphic functions on a strictly pseudoconvex domain withC ² boundary or a polydisc or a compact bordered Riemann surface or a bounded domainD such that intD = D is invertible if and only if it is a Fredholm operator if and only if is a holomorphic automorphism.     

Analytic continuation of holomorphic correspondences and equivalence of domains in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

The following result is proved: Let D and D′ be bounded domains in ℂ ⁿ , ∂D is smooth, real-analytic, simply connected, and ∂D′ is connected, smooth, real-algebraic. Then there exists a proper holomorphic correspondence f:D→D′ if and only if there exist points p∈∂D and p′∈∂D′, such that ∂D and ∂D′ are locally CR-equivalent near p and p′. This leads to a characterization of the equivalence relationship between bounded domains in ℂ ⁿ modulo proper holomorphic correspondences in terms of local CR-equivalence of their boundaries. Oblatum 23-I-2002 & 18-XI-2002?Published online: 17 February 2003     

Holomorphic functions which are highly nonintegrable at the boundary

Given a bounded convex domainD in ℂ^N with smooth boundary and a positive continuous function ϕ ond, it is proved that there is a holomorphic functionf onD such that |f|ϕ is nonintegrable onM∩D wheneverM is a real submanifold of a neighbourhood of a point ofbD which intersectsbD transversely.     

Proper holomorphic discs avoiding closed convex sets

Denote by the open unit disc in . Let C be a closed convex subset of . We prove that for each there is a proper holomorphic map such that and if and only if either C is a complex line or C does not contain any complex line. Received: 17 July 2001; in final form: 22 November 2001 / Published online: 5 September 2002     

Boundary structure of bounded symmetric domains

Let D be a bounded symmetric domain realized as the open unit ball of a complex Banach space E and denote for every by s _a the symmetry of D about a. We show that for every boundary point the locally uniform limit s _c := lim _{c → c} s _a exists as holomorphic map s _c :D→E and has values in the boundary of D. Received: 7 August 1999 / Revised version: 3 November 1999     

Regularity of discs attached to a submanifold ofC n

Letp be an analytic disc attached to a generating CR-submanifoldM of C ⁿ . It is proved that some recently introduced conditions onp andM which imply that the family of all smallC ^α holomorphic perturbations ofp alongM is a Banach submanifold of (A^α(D))ⁿ are equivalent. These conditions are given in terms of the partial indices of the discp attached toM and “holomorphic sections” of the conormal bundle ofM along p(∂D). Also, a sufficient geometric conditionon p andM is given so that the family of all smallC ^α holomorphic perturbationsof p alongM, fixed at some boundary point, is a Banach submanifold of (A ^α (D))ⁿ.     

Algebraicity of analytic maps to a hyperbolic variety

Let X be a complex algebraic variety. We say that X is Borel hyperbolic if, for every finite type reduced scheme S over the complex numbers, every holomorphic map from S to X is algebraic. We use a transcendental specialization technique to prove that X is Borel hyperbolic if and only if, for every smooth affine complex algebraic curve C, every holomorphic map from C to X is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity.     

Holomorphic Cliffordian functions

The aim of this paper is to put the foundations of a new theory of functions, called holomorphic Cliffordian, which should play an essential role in the generalization of holomorphic functions to higher dimensions. Let ℝ_0,2m+1 be the Clifford algebra of ℝ^2m+1 with a quadratic form of negative signature, be the usual operator for monogenic functions and Δ the ordinary Laplacian. The holomorphic Cliffordian functions are functionsf: ℝ^2m+2 → ℝ_0,2m+1, which are solutions ofDδ ^m f = 0. Here, we will study polynomial and singular solutions of this equation, we will obtain integral representation formulas and deduce the analogous of the Taylor and Laurent expansions for holomorphic Cliffordian functions. In a following paper, we will put the foundations of the Cliffordian elliptic function theory.     

A characterization of domains in <Emphasis Type="Bold">C</Emphasis><Superscript>2</Superscript> with noncompact automorphism group

Let D be a bounded domain in C ² with a non-compact group of holomorphic automorphisms. Model domains for D are obtained under the hypotheses that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type. The author was supported by DST (India) Grant No.: SR/S4/MS-283/05 and in part by a grant from UGC under DSA-SAP, Phase IV.     

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f ^(k)(z)| < |h(z)|; (b) f ^(k)(z) ≠ h(z). Then F is normal on D.