Non-constant Holomorphic Functions Research Articles

Spectra of Some Composition Operators

If H is a Hilbert space of holomorphic functions on the unit ball BN in CN and φ is a non-constant holomorphic map of the unit ball into itself, the composition operator Cφ is the operator on H defined by Cφf = f ∘ φ. In this paper, we give spectral information for bounded composition operators on some weighted Hardy spaces under the condition that φ is univalent and has a fixed point in the ball. When H is the usual Hardy space or a standard weighted Bergman space on the unit disk, this information shows that the spectrum of the composition operator is the disk centered at 0 whose radius is the essential spectral radius of the operator together with some isolated eigenvalues.

On the characterization of a Riemann surface by its semigroup of endomorphisms

Suppose D 1 {D_1} and D 2 {D_2} be Riemann surfaces which have bounded nonconstant holomorphic functions. Denote by E ( D i ) E({D_i}) , i = 1 , 2 i = 1,2 , the semigroups of all holomorphic endomorphisms. If ϕ : E ( D 1 ) → E ( D 2 ) \phi :E({D_1}) \to E({D_2}) is an isomorphism of semigroups then there exists a conformal or anticonformal isomorphism ψ : D 1 → D 2 \psi :{D_1} \to {D_2} such that ϕ \phi is the conjugation by ψ \psi . Also the semigroup of injective endomorphisms as well as some parabolic surfaces are considered.

Homogeneous line bundles over a toroidal group

A connected complex Lie group without non-constant holomorphic functions is called a toroidal group ([5]) or an (H, C)-group ([9]). Let X be an n-dimensional toroidal group. Since a toroidal group is commutative ([5], [9] and [10]), X is isomorphic to the quotient group Cn/Γ by a lattice of Cn. A complex torus is a compact toroidal group. Cousin first studied a non-compact toroidal group ([2]).Let L be a holomorphic line bundle over X. L is said to be homogeneous if is isomorphic to L for all x ε X, where Tx is the translation defined by x ε X. It is well-known that if X is a complex torus, then the following assertions are equivalent:

Homogeneous Complex Manifolds with more than One End

For homogeneous spaces of a (real) Lie group one of the fundamental results concerning ends (in the sense of Freudenthal [8] ) is due to A. Borel [6]. He showed that if X = G/H is the homogeneous space of a connected Lie group G by a closed connected subgroup H, then X has at most two ends. And if X does have two ends, then it is diffeomorphic to the product of R with the orbit of a maximal compact subgroup of G.In the setting of homogeneous complex manifolds the basic idea should be to find conditions which imply that the space has at most two ends and then, when the space has exactly two ends, to display the ends via bundles involving C* and compact homogeneous complex manifolds. An analytic condition which ensures that a homogeneous complex manifold X has at most two ends is that X have non-constant holomorphic functions and the structure of such a space with exactly two ends is determined, namely, it fibers over an affine homogeneous cone with its vertex removed with the fiber being compact [9], [13].

An extremal problem for Dirichlet-finite holomorphic functions on Riemann surfaces

Iff is a nonconstant holomorphic function with finite Dirichlet integralD(f) on a Riemann surfaceR, then |f|2 has the least harmonic majorantf 2 onR. We show Σf 2(a ≦π −1 D(f)), wherea runs over all the roots off = 0 onR. The equality holds if and only iff is of type ℬl1 fromR onto a disk of center 0. A consideration is proposed for the non-Euclidean case.

On Weakly 1-Complete Surfaces without Non-Constant Holomorphic Functions

In this paper, we are interested in a class of two dimensional complex manifolds which are called weakly 1-complete surfaces. Here we call a two dimensional complex manifold X a weakly 1-complete surface if X possesses a C°°-exhausting plurisubharmonic function. This class includes two different extreme objects: compact analytic surfaces and two dimensional Stein manifolds. But at the same time, this class includes some curious examples from the function theoretic point of view i.e. there are weakly 1-complete surfaces without non-constant holomorphic functions (see [3] [6] [8] [12]) and moreover non-compact weakly 1-complete surfaces have an extreme function theoretic property i.e. a non-compact weakly 1-complete surface X is holomorphically convex if and only if X possesses a non-constant holomorphic function (see [9]). Looking back to the case of compact analytic surfaces, roughly speaking, they are classified by the existence or non-existence of meromorphic function. Hence it is natural to suppose that this aspect might give a new standpoint to analyze such a curious example in the class of weakly 1-complete surfaces as far as weakly 1-completeness is expected as a nice intermediate concept between compactness and Stein. This note is an attempt towards the problem of the existence of meromorphic function on non-compact weakly 1-complete surfaces. From now on, all weakly 1-complete surfaces are connected and non-compact and have no exceptional compact curves of the first kind unless otherwise is explicitly stated. Then we shall prove the following theorem.

Ends of complex homogeneous manifolds having non-constant holomorphic functions

Ends of complex homogeneous manifolds having non-constant holomorphic functions

Weakly 1-Complete Manifold and Levi Problem

Let X be a paracompact complex manifold of dimension «J>1. We call X a weakly 1-complete manifold if there exists a C°° plurisubharmonic function (? on X such that for every ceR (real number), <p~1((— oo? c)} is relatively compact in X. cp is called an exhaustion function of X. It is well known that holomorphically convex manifolds are weakly 1-complete. The converse is not true in general (cf. [1]), so one is led to the problem of seeking natural additional conditions which make weakly 1-complete manifolds holomorphically convex. The content of this article is divided into two parts; Section 1 is devoted to prove some properties of weakly 1-complete manifolds which have a nonconstant holomorphic function. In Section 2, first we present an application of the Nakano's vanishing theorem to the Levi problem on projective spaces and hyperquadrics. These results are not new (cf. [3]) but the method will be of some interest. Next, combining the Nakano's vanishing theorem with the result in Section 1 and a well known theorem (due to Bonnet) of differential geometry we obtain the following.

On non-compact complex nil-manifolds

Given a complex manifold X, it is natural to consider the quotient space X~ ~, where p~q whenever f(p)=f(q) for every fe(9(X). In the best of all worlds one would hope that X / ~ is a Stein space, and that there are no non-constant holomorphic functions on the equivalence classes of ~ . One would also hope to explain the latter phenomenon by compactness or Levi curvature. A classical result of this type is the Remmert Reduction Theorem: If X is an irreducible, holomorphically convex analytic space, then X / ~ is a Stein analytic space and the natural holomorphic separation map F : X ~ X / ~ is proper [15]. In general it is not possible to prove any such reduction theorem. In fact there are naturally occuring manifolds X where there is no natural way to put a complex structure on X~ ~ [5]. Grauert's examples [5] all contain non-compact divisors on which every holomorphic function is constant. Off these divisors the holomorphic functions separate points and give local coordinates. If one assumes that X is homogeneous, then of course X can't be one of Grauert's examples. For the purposes of this paper, homogeneity means that a complex Lie group G acts holomorphically and transitively on X. In other words X is holomorphically equivalent to G/H, where G is a connected complex Lie group and H is a closed subgroup of G. One would hope to obtain some form of a reduction theorem in the homogeneous setting. It is true that the holomorphic separation map of a complex homogeneous manifold can always be realized as a holomorphic fibration over a holomorphically separable homogeneous manifold. However, we do not even know if a holomorphically separable homogeneous manifold always has an envelope of holomorphy. Clearly it need not be Stein, e.g. ~2\{(0,0)}. Also the fiber of the holomorphic separation map may have nonconstant hotomorphic functions (see [2]). In particular cases, some results have been obtained. Morimoto [13] showed that if G is a connected, complex Lie group, then it has a closed, connected, central,

Families of Holomorphic Maps into Riemann Surfaces

In analogy with the Hartogs theorem that separate analyticity of a function implies analyticity, it is shown that a separately normal family of holomorphic maps from a polydisk into a Riemann surface is a normal family. This contrasts with examples of discontinuous separately analytic maps from a bidisk into the Riemann sphere. The proof uses a theorem on pseudoconvexity of normality domains, which is proved via the following convergence criterion: a sequence $\{ {f_j}\}$ of holomorphic maps from a complex manifold into a Riemann surface converges to a nonconstant holomorphic map if and only if the sequence $\{ f_j^{ - 1}\}$ of set-valued maps, defined on the Riemann surface, converges to a suitable set-valued map. Extending Osgood’s theorem, it is also shown that a separately analytic map (resp. a separately normal family of holomorphic maps) from a polydisk into a hyperbolic complex space is analytic (resp. normal).

Neighborhoods of a Compact Non-Singular Algebraic Curve Imbedded in a 2-Dimensional Complex Manifold

Let C be a compact non-singular algebraic curve imbedded in a 2-dimensional complex manifold S. In this paper we consider the following problem: Under what conditions does there exist a non-constant holomorphic function defined on a small neighborhood of Cl The signature of the normal bundle Nc is not sufficient to solve our problem (see, Table in § 6 and Theorem 2 in § 5). Hence we have to introduce the concept of a regularly half pseudoconvex neighborhood system of C (for definition, see (1.1)). Then the necessary and sufficient condition is given in the following

Families of holomorphic maps into Riemann surfaces

In analogy with the Hartogs theorem that separate analyticity of a function implies analyticity, it is shown that a separately normal family of holomorphic maps from a polydisk into a Riemann surface is a normal family. This contrasts with examples of discontinuous separately analytic maps from a bidisk into the Riemann sphere. The proof uses a theorem on pseudoconvexity of normality domains, which is proved via the following convergence criterion: a sequence { f j } \{ {f_j}\} of holomorphic maps from a complex manifold into a Riemann surface converges to a nonconstant holomorphic map if and only if the sequence { f j − 1 } \{ f_j^{ - 1}\} of set-valued maps, defined on the Riemann surface, converges to a suitable set-valued map. Extending Osgood’s theorem, it is also shown that a separately analytic map (resp. a separately normal family of holomorphic maps) from a polydisk into a hyperbolic complex space is analytic (resp. normal).

Integrals and derivatives of functions in MacLane's class and of normal functions

If f is meromorphic in D = {,|z| &lt; 1}, we say that f has the asymptotic value a at ζ, ζ ∈ C(={|z| = 1}), if there exists an arc Γ such that Γ ends at ζ, (Γ − {ζ}) ⊂ D, and f has the limit a as |z| → 1 on Γ. The class (M) originally defined by G. R. MacLane ((8), p. 8), consists of those functions that are non-constant and holomorphic (meromorphic) in D and that have asymptotic values at a dense subset of C. In (8), p. 36, MacLane has shown the following three conditions are sufficient for a non-constant holomorphic function f to be in :

Normal families and the Nevanlinna theory

1. Let ~ be a family of nonconstant holomorphic functions defined in the disc A = {Izl < 1}. :~ is said to be normal if every sequence of functions in :~ either contains a subuniformly convergent subsequence, or contains a subsequence which converges subuniformly to the constant co. A family :~ of meromorphic functions is normal when every sequence of functions of :~ has a subsequence which is subuniformly convergent with respect to the chordal metric. P. Montel [15] first realized the scope and coherence of these families, and used them to give a particularly unified t reatment of Picard's great theorems, and Schottky's and Landau's theorems. The fact that these results were so intimately related led A. Bloch to the hypothesis tha t precisely those properties which reduce a function meromorphic in C (= { I z I < ~ }) to a constant, make normal a family of functions meromorphic in A.

Composition Operators

The object of this note is to report on some of the properties of a class of operators induced by inner functions. If m is normalized Lebesgue measure on the unit circle X in the complex plane and Cϕ is an inner function (a complex function on X of unit modulus almost everywhere whose Poisson integral is a non-constant holomorphic function in the open unit disk), then an operator Cϕ on L2(m) is defined by