Geometric Function Theory in Several Complex Variables

Preface.

Contents.

No abstract received.

The greatest computed value of π(x), the number of primes not exceed x, according to 3, p.3, 4, p.178, is.

(by Lagarias, Miller & Odlyzko in 1985, in Bell Lab.), which is also the known result up to the best of the author’s knowledge. Recently we obtain a revised version of combinatory algorithm A in 5, p.180 (we refer to as Meissel–Lehmer– Lagarias–Odlyzko’s Algorithm) so as to attain a new program of finding π(x) in O(x^1/4+^∈) space and in O(x⁵/^8+∈) + O(x^2/3/logx) time. (∈ being arbitrary small positive number.) This program will enable us to compute individual value of π(x) for any x, say, x ≤ 10²¹, within reasonable time and space.

In this paper, we show that some of the function-theoretic properties of Arveson’s Hardy space in several complex variables to the infinite complex variables. The connection is clarified between operator theory of contractions for infinite sequences of mutually commuting operators on a Hilbert space and function theory of holomorphic functions in infinite complex variables. Finally, von Neumann’s inequality for infinite sequences of mutually commuting operators is proved.

In this paper, it is proved that the (radially) boundary behavior of pluriharmonic functions in the unit ball of Cⁿ obeys, in some sense, the law of the iterated logarithm. The main ingredients in the proof are a distribution function inequality for Littlewood-Paley g_*-functions and Brownian motion in the unit ball of Cⁿ.

In this paper, we characterize the existence of proper holomorphic mapping between a kind of nonsmooth Reinhardt domains defined as.

here p = (p₁, … , p_n) ∈ (R+)ⁿ, q = (q₁, … , q_m) ∈ (R⁺)^m; n > 1, m > 1.

In this paper we study biholomorphic mappings on the unit ball in ℂⁿ which are spirallike with respect to a boundary point. To do this we investigate semigroups of holomorphic self-mappings of the unit ball having boundary fixed points. In particular, we establish an infinitesimal analog of the Julia–Carathéodory Theorem.

Fix a normalized, univalent function ƒ in the unit disk. Consider a Moebius mapping of the unit disk into itself. Then apply the original function ƒ. Renormalize so that the origin goes to the origin and the derivative at the origin is one. In most cases, this transformation gives a new function. For certain functions ƒ and certain Moebius transformations the function does not change. These invariant functions include solutions to many extremal problems. A similar invariance problem is studied in the unit ball in two complex variables.

The principal fact in complex analysis is the Cauchy theorem, which states that, if a domain Ω in the complex plane is rectifiably bounded by finitely many Jordan curves, then the boundary integral of each function holomorphic on Ω and continuous on its closure vanishes. In fact, the words ‘finitely many’ are superfluous.

No abstract received.

We show the existence of a biholomorphic map σ, defined near 0 = σ(0) ∈ C² and satisfying the following: (a) eigenvalues of σ are not root of unity, σ preserves the standard holomorphic 2-form dξ ∧ dη, and σ is reversible by an anti-holomorphic involution and by a formally holomorphic involution; (b) σ is not reversible by any C¹-smooth involution with a holomorphic linear part; in particular, σ is not reversible by any holomorphic involution. The proof is obtained by showing the absence of certain periodic orbits of σ.

In this paper we survey some of the most important properties of Loewner chains and their transition mappings on the unit ball B of ℂⁿ with respect to an arbitrary norm. We begin with a discussion of the class of holomorphic mappings from B into ℂⁿ which plays the role of the Carathéodory class in higher dimensions. We consider the Lipschitz regularity in t of an arbitrary Loewner chain, and in s and t of the associated transition mapping v(z, s, t), improving known results. Various consequences are given. Next we consider the role of parametric representation in Loewner theory in higher dimensions. We show that the class S⁰(B) of mappings which have parametric representation on B has some properties in common with the class S of normalized univalent functions on the unit disc. For example, a mapping f in the set H(B) of holomorphic mappings from B into ℂⁿ has parametric representation if and only if there exists a Loewner chain f(z, t) such that the family {e^−tf(z, t)}_t≥0 is a normal family on B and f(z) = f(z, 0) for z ∈ B. Also, the set S⁰(B) is compact as a subset of H(B). Finally we consider some applications to the Roper-Suffridge extension operator.

The definition and properties of the Euler-Lagrange cohomology groups , 1 ≤ k ≤ n, on a symplectic manifold (M²ⁿ, ω) are given and studied. For k = 1 and k = n, they are isomorphic to the corresponding de Rham cohomology groups and , respectively. The other Euler-Lagrange cohomology groups are different from either the de Rham cohomology groups or the ones with respect to the harmonic forms on (M²ⁿ, ω), in general. The general volume-preserving equations on (M²ⁿ, ω) are also presented from cohomological point of view. In the special cases, these equations become the ordinary Hamiltonian equations.

This paper gives a new inequality. Let the vector x = (x₁, x₂, … , x_n) and denote . If . Then we have , where and ω_n,_p(a, b, c, e) is a positive function of numbers n, p and vectors a, b, c, e. And some important applications of this inequality are given.

We define an extended Cesàro operator T_g with holomorphic symbol g in the unit ball B of Cⁿ as.

Where is the radial derivative of g. In this paper we characterize those g for which T_g is bounded (or compact) on the Bloch space and the little Bloch space .

No abstract received.

In this paper, by using the Plemelj formula with the Cauchy principal value and the Hadamard principal value of higher order mixed partial derivatives for integral of the Bochner-Martinelli type on a closed smooth orientable manifold in Cⁿ, we obtain the composite formula of higher order singular integral for the Bochner-Martinelli kernel. Moreover, we successfully discuss the variable coefficient linear higher order partial differential integral equation on a closed smooth manifold, and obtain its regularization theorem.

Let Ω be a bounded balanced pseudoconvex domain in Cⁿ that admits a plurisubhamonic defining function of class C². We prove that if ƒ is a spirallike mapping of type α on Ω, then ƒ can be expressed a limiting form concerning a Schwarz mapping, Moreover, the characteristics for starlike mappings and spirallike mappings of type α are obtained. As a direct application, the growth theorem for spirallike mappings of type α is set up.

A class of biholomorphic mappings named quasi-convex mapping is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings and contains the class of convex mappings properly, and it has the same growth and covering theorems as the convex mappings. Furthermore, when the Banach space is confined to ℂⁿ, the quasi-convex mapping is exactly the quasi-convex mapping of type A introduced by K.A. Roper and T.J. Suffridge.

No abstract received.

By means of the invariant integral kernel (Demailly and Laurent-Thiebaut kernel), complex Finsler metric and non-linear connection associated with Chern-Finsler connection to research integral representation theory of complex Finsler manifolds, the Koppelman formula is obtained.

Many results of geometric function theory in the complex plane (i.e. in dimension one) have extensions to higher dimensions with some surprising differences and difficulties. We survey some of the known results including the dependence of the theory on the specific domain. A major emphasis will be on convex mappings of the Euclidean ball and questions concerning boundary behavior.

No abstract received.

By introducing a pseudo-distance function on compact Riemann surfaces, we give a new type of Hadamard theorem.

In this paper, we define a Hodge-Laplace operator on a compact complex Finsler manifolds, it descends to the usual Hodge Laplace operator on Hermitian manifold when the manifold is Hermitian. The key point of defining this Hodge-Laplace operator is to define a global inner product on the compact complex Finsler manifold. we do this by pulling the differential forms of type-(p, q) on M back to the projectivized tangent bundle ℙTM of M and then using the natural Hermitian inner product on ℙTM to obtain a global inner product on M.

Let ϕ(z) = (ϕ₁(z), … ,ϕⁿ(z)) be a holomorphic self-map of ⅅⁿ and ψ(z) a holomorphic function on ⅅⁿ, where ⅅⁿ is the unit polydiscs of ℂⁿ. Let 0 < α < 1, this paper gives some necessary and sufficient conditions for the weighted composition operator W_ψ,ϕ induced by ψ and ϕ to be bounded and compact on the Lipschitz space Lip_α(ⅅⁿ).