On a class of holomorphic functions representable by Carleman formulas in the interior of an equilateral cone from their values on its rigid base

Let Δ be an equilateral cone in C with vertices at the complex numbers and rigid base M (Section 1). Assume that the positive real semi-axis is the bisectrix of the angle at the origin. For the base M of the cone Δ we derive a Carleman formula representing all those holomorphic functions from their boundary values (if they exist) on M which belong to the class . The class is the class of holomorphic functions in Δ which belong to the Hardy class near the base M (Section 2). As an application of the above characterization, an important result is an extension theorem for a function fL¹(M) to a function .     

Applications holomorphes propres entre certains domaines bornés de ℂn

In this Note, we developp a new technic to study the existence of proper holomorphic mappings between a strictly pseudoconvex domain and certain special non-regular domains in ℂⁿ. In particular, it can be applied in the case of the minimal ball and the Lie ball. We prove that self-proper holomorphic mappings between such domains are biholomorphic. Furthermore, we establish a necessary and sufficient condition to factorize a proper holomorphic mapping by automorphisms. Scaling method is applied for the first time in a singular points of the boundary of the domains.     

Proper holomorphic mappings and related automorphism groups

In this note we determine the automorphism group of complex manifolds which are proper images of a simply connected strictly pseudoconvex domain in ?ⁿ. We also investigate automorphisms of domains invariant under a compact subgroup of complex linear transformations. Furthermore, some regularity and rigidity properties of proper holomorphic mappings are established. In particular we solve a question raised by Hahn and Pflug regarding the nonexistence of proper holomorphic mappings between the euclidian ball and the complex minimal ball of ?ⁿ.     

Complex<Emphasis Type="BoldItalic">n</Emphasis>-dimensional manifolds with a real<Emphasis Type="BoldItalic">n</Emphasis><Superscript>2</Superscript>-dimensional automorphism group

The main theorem of this article is a characterization of non compact simply connected complete Kobayashi hyperbolic complex manifold of dimension n≽ 2 with real n ²-dimensional holomorphic automorphism group. Together with the earlier work [11, 12] and [13] of Isaev and Krantz, this yields a complete classification of the simply-connected, complete Kobayashi hyperbolic manifolds with dim_ℝ Aut (M) ≽ (dim_ℂ M)².     

Extension Operators Preserving Biholomorphic Mappings on Hartogs Domains*

In this paper, the authors extend the Roper-Suffridge operator on the generalized Hartogs domains. They mainly research the properties of the extended operator.By the characteristics of Hartogs domains and the geometric properties of subclasses of spirallike mappings, they obtain the extended Roper-Suffridge operator preserving almost starlikeness of complex order λ, almost spirallikeness of type β and order α, parabolic spirallikeness of type β and order ρ on the Hartogs domains in different co...     

A new Lichnerowicz–Obata estimate in the presence of a parallel p-form

 Let (M ⁿ ,g) be a compact Riemannian manifold with a smooth boundary. In this paper, we give a Lichnerowicz-Obata type lower bound for the first eigenvalue of the Laplacian of (M ⁿ ,g) when M has a parallel p-form (2 ≤p≤n/2). This result follows from a new Bochner-Reilly's formula. Moreover, we give a characterization of the equality case when (M ⁿ ,g) is simply connected. Received: 1 June 2001     

Proper holomorphic mappings between symmetrized ellipsoids

We characterize the existence of proper holomorphic mappings in the special class of bounded (1, 2, . . . , n)-balanced domains in \mathbb Cⁿ,{\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 groups of these domains.     

Proper holomorphic self-maps of smooth bounded Reinhardt domains in ℂn

It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain of D’Angelo finite type in ℂⁿ (n1) is an automorphism.

     

A note on K<Emphasis Type="Italic">ä</Emphasis>hler manifolds with almost nonnegative bisectional curvature

In this note, we prove the following result. There is a positive constant ε(n, Λ) such that if M ⁿ is a simply connected compact Kähler manifold with sectional curvature bounded from above by Λ, diameter bounded from above by 1, and with holomorphic bisectional curvature H ≥ ?ε(n, Λ), then M ⁿ is diffeomorphic to the product M ₁ × ? × M _k , where each M _i is either a complex projective space or an irreducible Kähler–Hermitian symmetric space of rank ≥ 2. This resolves a conjecture of Fang under the additional upper bound restrictions on sectional curvature and diameter.     

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     

Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S¹ ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.     

A Remark on a Theorem by Kodama and Shimizu

We prove a characterization theorem for the unit polydisc Δ ⁿ ⊂ℂ ⁿ in the spirit of a recent result due to Kodama and Shimizu. We show that if M is a connected n-dimensional complex manifold such that (i) the group Aut (M) of holomorphic automorphisms of M acts on M with compact isotropy subgroups, and (ii) Aut (M) and Aut (Δ ⁿ ) are isomorphic as topological groups equipped with the compact-open topology, then M is holomorphically equivalent to Δ ⁿ .      

Holomorphic Lefschetz formula for manifolds with boundary

The classical Lefschetz fixed point formula expresses the number of fixed points of a continuous map f:MM in terms of the transformation induced by f on the cohomology of M. In 1966 Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they presented a holomorphic Lefschetz formula for compact complex manifolds without boundary, a result, in the framework of algebraic geometry due to Eichler (1957) for holomorphic curves. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, hence the Atiyah-Bott theory is no longer applicable. To get rid of the difficulties related to the boundary behaviour of the Dolbeault cohomology, Donelli and Fefferman (1986) derived a fixed point formula for the Bergman metric. The purpose of this paper is to present a holomorphic Lefschetz formula on a strictly convex domain in ⁿ, n>1.Mathematics Subject Classification (2000):32S50; 58J20*Supported by the Deutsche Forschungsgemeinschaft and the RFFI grant 02–01–00167.**Supported by the Deutsche Forschungsgemeinschaft and the RFFI grant 02–01–00167.     

Complète intégrabilité singulière

We show that an holomorphic vector field in a neighbourhood of its singular point 0 0 ∈ ℂⁿ is analytically normalizable as soon as it has a sufficiently large number of commuting holomorphic vector fields, a sufficiently large number of formal first integrals, and that a Diophantian small divisors condition related to its linear part is satisfied.     

Steinness of the Total Space of a Non‐Trivial Algebraic Affine ℂ‐Bundle on the Punctured Complex Affine Plane

We prove that the total space E of an algebraic affine ℂ‐bundle π : E → X on the punctured complex affine plane X ≔ ℂ² – {(0, 0)} is Stein if and only if it is not isomorphic to the trivial holomorphic line bundle X × ℂ.     

Conformally flat,minimal, Lagrangian submanifolds in complex space forms

We investigate n-dimensional (n ⩾ 4), conformally flat, minimal, Lagrangian submanifolds of the n-dimensional complex space form in terms of the multiplicities of the eigenvalues of the Schouten tensor and classify those that admit at most one eigenvalue of multiplicity one. In the case where the ambient space is ℂⁿ, the quasi umbilical case was studied in Blair (2007). However, the classification there is not complete and several examples are missing. Here, we complete (and extend) the classification and we also deal with the case where the ambient complex space form has non-vanishing holomorphic sectional curvature.