复Banach空间中C-R方程的全纯解

二重复数是复数的一种推广，在其上的全纯映照族对应于Ｃ２上满足复Ｃａｕｃｈｙ－Ｒｉｅｍａｎｎ方程的全纯映照族．可以证明，这样的映照族本质上是由二个单复变数的全纯函数的直乘积所组成的族．本文证明：即使在Ｂａｎａｃｈ空间中，Ｃａｕｃｈｙ－Ｒｉｅｍａｎｎ方程的全纯解，具有同样的性质．     

Sets of Approximation and Interpolation in ℂ for Manifold-Valued Maps

We give examples of non-smooth sets in the complex plane with the property that every holomorphic map continuous to the boundary from these sets into any complex manifold may be uniformly approximated by maps holomorphic in some neighborhood of the set (Mergelyan-type approximation for manifold-valued maps.) Similar results are proved for sections of complex-valued holomorphic submersions from complex manifolds.      

On holomorphic immersions into K(a)hler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f: X → M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(X) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f)measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form.In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.     

Harnack inequalities, Kobayashi distances and holomorphic motions

We prove some generalizations and analogs of the Harnack inequalities for pluriharmonic, holomorphic and “almost holomorphic” functions. The results are applied to proving smoothness properties of holomorphic motions over almost complex manifolds.     

HOLOMORPHIC HARMONIC ANALYSIS ON COMPLEX REDUCTIVE GROUPS

The authors define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties such as the Fourier inversion formula, and give some applications. The definition of the holomorphic Fourier transform makes use of the notion of K-admissible measures. The authors prove that K-admissible measures are abundant, and the definition of holomorphic Fourier transform is independent of the choice of K-admissible measures.     

Some Results of Holomorphic Vector Bundles on Non-primary Hopf Manifolds

In recent years,there are a lot of work on holomorphic vector bundles on non-algebraic     

An extension of Calabi's rigidity theorem to complex submanifolds of indefinite complex space forms

The rigidity for full holomorphic isometric immersions of an indefinite Kähler manifold into an indefinite complex space form is proved. All such immersions between indefinite complex projective (and hyperbolic) spaces are founded and examples of non-congruents holomorphic isometric immersions are exposed.     

Relative connections on principal bundles and relative equivariant structures

We investigate relative holomorphic connections on a principal bundle over a family of compact complex manifolds. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic principal bundle over a complex analytic family. We also introduce the notion of relative equivariant bundles and establish its relation with relative holomorphic connections on principal bundles.     

Representation of holomorphic functions of many variables by Cauchy-Stieltjes-type integrals

We consider functions of many complex variables that are holomorphic in a polydisk or in the upper half-plane. We give necessary and sufficient conditions under which a holomorphic function is a Cauchy-Stieltjes-type integral of a complex charge. We present several applications of this criterion to integral representations of certain classes of holomorphic functions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 522–542, April, 2006.     

Holomorphic extensions and theta functions on complex tori

In this paper we give a complete characterization of vector bundles of any dimension over complex tori in which the Hartogs–Bochner holomorphic extension phenomenon holds. Since holomorphic sections of line bundles over complex tori can be identified with theta functions, the results are formulated in terms of this class.     

On the holomorphic sectional and bisectional curvatures in complex Finsler geometry

The concepts of holomorphic sectional and bisectional curvatures for holomorphic vector bundles in complex Finsler geometry are used to characterize the concept of big vector bundles in algebraic geometry.     

Spherical shells as obstructions for the extension of holomorphic mappings

In this paper we study the extension properties of holomorphic and meromorphic maps into complex manifolds that carry a pluriclosed Hermitian metric. For example, any compact, complex surface admits such a metric. We prove that the only obstructions for the Hartogs-type extendability of holomorphic maps are spherical shells and rational curves.     

Über analytisch abhängige holomorphe und meromobphe Abbildungen

We discuss the conceptions of analytic dependence, strict analytic dependence, spaces of level sets and complex bases of holomorphic mappings and their relationship. For instance, a theorem of the following type is proved: Let X be a normal complex space, let be holomorphic mappings, which are analytically related (i.e. depends analytically on and depends analytically on). Let be maximal (i.e. majorizes every holomorphic map which is strictly analytically dependent on) and let be strongly maximal (i.e. majorizes every holomorphic map which depends analytically on). Then Z^* is a proper modification of, if is proper or nowhere degenerated.     

Totally geodesic holomorphic subspaces

In [G. Munteanu, Complex Spaces in Finsler, Lagrange and Hamilton Geometries, vol. 141, Kluwer Academic Publishers, Dordrecht, FTPH, 2004.] we underlined the motifs of a remarkable class of complex Finsler subspaces, namely the holomorphic subspaces. With respect to the Chern–Finsler complex connection (see [M. Abate, G. Patrizio, Finsler Metrics—A Global Approach, Lecture Notes in Mathematics, vol. 1591, Springer, Berlin, 1994.]) we studied in [G. Munteanu, The equations of a holomorphic subspace in a complex Finsler space, Publicationes Math. Debrecen, submitted for publication.] the Gauss, Codazzi and Ricci equations of a holomorphic subspace, the aim being to determine the interrelation between the holomorphic sectional curvature of the Chern–Finsler connection and that of its induced tangent connection.In the present paper, by means of the complex Berwald connection, we study totally geodesic holomorphic subspaces. With respect to complex Berwald connection the equations of the holomorphic subspace have simplified expressions. The totally geodesic subspace request is characterized by using the second fundamental form of complex Berwald connection.     

On Biholomorphy in Infinite Dimensions

In this article, we consider the question of when a holomorphic mapping on a domain in a complex Hilbert space into itself has a holomorphic inverse. We give a strengthened version of a known result that involves a Fredholm condition on the mapping. We show that holomorphic mappings on certain domains that are biholomorphic near the boundary are biholomorphic on the domain itself.      

The holomorphic automorphism group of the complex disk

The group of all holomorphic automorphisms of the complex unit disk consists of Möbius transformations involving translation-like holomorphic automorphisms and rotations. The former are calledgyrotranslations. As opposed to translations of the complex Plane, which are associative-commutative operations forming a group, gyrotranslations of the complex unit disk fail to form a group. Rather, left gyrotranslations are gyroassociative-gyrocommutative operations forming agyrogroup.     

多复变数全纯映射精细的Bohr定理

本文首先建立不依赖自同构从复Banach空间平衡域到Cn单位多圆柱上一定限制条件下全纯映射精细的范数型Bohr定理及复Banach空间X上单位球到复Banach空间Y上单位球全纯映射精细的泛函型Bohr定理.其次,给出有界对称域上全纯映射精细的Bohr定理.最后,得到J*代数单位球上全纯映射精细的Bohr定理.所得结果将一维的Bohr定理推广至高维.     

Complex planar splines

In contradistinction to the known theory on complex splines which are defined on the boundary of a region in , we define complex planar splines on a region itself as a complex valued continuous function which is defined piecewise on suitable meshes of that region. The main idea is to use nonholomorphic functions as pieces, since holomorphic pieces would lead to just one holomorphic function on the whole region. Some of the techniques used are available from the theory of finite elements. But we also consider new aspects, namely, mapping properties of a complex planar spline v and the difference f − v, where f is, in general, a holomorphic function. For triangular meshes, rectangular and parallelogrammatic meshes, and meshes on circular sectors, explicit expressions are provided; also properties of the newly introduced complex planar splines are studied.     

On the relative connections

Abstract

We investigate relative connections on a sheaf of modules. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic vector bundle over a complex analytic family. We show that the relative Chern classes of a holomorphic vector bundle admitting relative holomorphic connection vanish, if each of the fiber of the complex analytic family is compact and Kähler.     

On the holomorphic torus-Bott tower of aspherical manifolds

We introduce a notion of holomorphic torus-Bott tower which is an iterated holomorphic Seifert fiber space with fiber a complex torus. This is thought of as a holomorphic version of a real Bott tower. The top space of the holomorphic torus-Bott tower is called a holomorphic torus-Bott manifold. We discuss the structure of holomorphic torus-Bott manifolds and particularly the holomorphic rigidity of holomorphic torus-Bott manifolds.