Complex Banach Manifold Research Articles

Aron–Berner–type extension in complex Banach manifolds

Let S S be a compact Hausdorff space and X X a complex manifold. We consider the space C ( S , X ) C(S,X) of continuous maps S → X S\to X , and prove that any bounded holomorphic function on this space can be continued to a holomorphic function, possibly multivalued, on a larger space B ( S , X ) B(S,X) of Borel maps. As an application we prove two theorems about bounded holomorphic functions on C ( S , X ) C(S,X) , one reminiscent of the Monodromy Theorem, the other of Liouville’s Theorem.

The $p$-integrable Teichmüller space for $p \geqslant 1$

We verify that the $p$-integrable Teichmüller space $T_{p}$ admits the canonical complex Banach manifold structure for any $p \geq 1$. Moreover, we characterize a quasisymmetric homeomorphism corresponding to an element of $T_{p}$ in terms of the $p$-Besov space for any $p>1$.

The VMO-Teichmüller space and the variant of Beurling–Ahlfors extension by heat kernel

We give a real-analytic section for the Teichmüller projection onto the VMO-Teichmüller space by using the variant of Beurling–Ahlfors extension by heat kernel introduced by Fefferman et al. (Ann Math 134:65–124, 1991). Based on this result, we prove that the VMO-Teichmüller space can be endowed with a real Banach manifold structure that is real-analytically equivalent to its complex Banach manifold structure. We also obtain that the VMO-Teichmüller space admits a real-analytic contraction mapping.

VMO-Teichmüller space on the real line

An increasing homeomorphism \(h\) on the real line \(\mathbb{R}\) is said to be strongly symmetric if it can be extended to a quasiconformal homeomorphism of the upper half plane \(\mathbb{U}\) onto itself whose Beltrami coefficient \(\mu\) induces a vanishing Carleson measure \(|\mu(z)|^2/y\,dx\,dy\) on \(\mathbb{U}\). We will deal with the class of strongly symmetric homeomorphisms on the real line and its Teichmüller space, which we call the VMO-Teichmüller space. In particular, we will show that if \(h\) is strongly symmetric on the real line, then it is strongly quasisymmetric such that \(\log h'\) is a VMO function. This improves some classical results of Carleson (1967) and Anderson-Becker-Lesley (1988) on the problem about the local absolute continuity of a quasisymmetric homeomorphism in terms of the Beltrami coefficient of a quasiconformal extension. We will also discuss various models of the VMO-Teichmüller space and endow it with a complex Banach manifold structure via the standard Bers embedding.

Teichmüller spaces of generalized symmetric homeomorphisms

We introduce the concept of a new kind of symmetric homeomorphism on the unit circle, which is derived from the generalization of symmetric homeomorphisms on the real line. By the investigation of the barycentric extension for this class of circle homeomorphisms and the biholomorphic automorphisms induced by trivial Beltrami coefficients, we show that the Bers Schwarzian derivative map is a holomorphic split submersion and endow a complex Banach manifold structure on the Teichmüller space of those generalized symmetric homeomorphisms.

Teichmüller space of circle diffeomorphisms with Hölder continuous derivatives

Based on the quasiconformal theory of the universal Teichmüller space, we introduce the Teichmüller space of diffeomorphisms of the unit circle with \alpha -Hölder continuous derivatives as a subspace of the universal Teichmüller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the Bers embedding. Then, we provide a complex Banach manifold structure for it and prove that its topology coincides with the one induced by local C^{1+\alpha} -topology at the base point.

Multiplicity-freeness of Unitary Representations in Sections of Holomorphic Hilbert Bundles

Abstract We prove several results asserting that the action of a Banach–Lie group on Hilbert spaces of holomorphic sections of a holomorphic Hilbert space bundle over a complex Banach manifold is multiplicity-free. These results require the existence of compatible anti-holomorphic bundle maps and certain multiplicity-freeness assumptions for stabilizer groups. For the group action on the base, the notion of an $(S,\sigma )$-weakly visible action (generalizing T. Koboyashi’s visible actions) provides an effective way to express the assumptions in an economical fashion. In particular, we derive a version for group actions on homogeneous bundles for larger groups. We illustrate these general results by several examples related to operator groups and von Neumann algebras.

On symmetric homeomorphisms on the real line

We introduce a complex Banach manifold structure on the space of normalized symmetric homeomorphisms on the real line.

Asymptotic Teichmüller space of a closed set of the Riemann sphere

The asymptotic Teichmüller space A T ( E ) AT(E) of a closed subset E E of the Riemann sphere C ^ \hat {\mathbb {C}} with at least 4 4 points and the natural asymptotic Teichmüller metric are introduced. It is proved that A T ( E ) AT(E) is isometrically isomorphic to the product space of the asymptotic Teichmüller spaces of the connected components of C ^ ∖ E \hat {\mathbb {C}}\setminus E and the Banach space of the Beltrami coefficients defined on E E . Furthermore, it is proved that there is a complex Banach manifold structure on A T ( E ) AT(E) .

Holomorphic contractibility and other properties of the Weil-Petersson and VMOA Teichmüller spaces

The Weil-Petersson and VMOA Teichmuller spaces, subspaces of the universal Teichmuller space, are complex Banach manifolds modelled on different Banach spaces. We show that they are holomorphically contractible. It is given in (28) that the Kobayashi and Teichmul- ler metrics coincide with each other on the Weil-Petersson Teichmuller space. We show that this property is also true on the VMOA Teichmuller space. A couple of other properties of the two spaces are also obtained.

Analytic cohomology groups of infinite dimensional complex manifolds

Given a cohesive sheaf S over a complex Banach manifold M, we endow the cohomology groups Hq(M,S) of M and Hq(U,S) of open covers U of M with a locally convex topology. Under certain assumptions we prove that the canonical map Hq(U,S)→Hq(M,S) is an isomorphism of topological vector spaces.

Old and New Invariant Pseudo-Distances Defined by Pluriharmonic Functions

We construct a new invariant pseudo-distance on a complex Banach manifold using the set of pluriharmonic functions with values in a proper open interval of \(\mathbb R\).

On Runge approximation in a Banach space

We show that on some open sets, more general than balls, Runge approximation is possible in certain Banach spaces, and also in certain complex Banach manifolds. We also show that there is an entire holomorphic curve in Hilbert space on which there is a bounded holomorphic function on the trace of a ball that has no bounded holomorphic extension to even a smaller concentric ball. Using the same technique we also prove that a form of Runge approximation better than an error function is not always possible.

Plurisubharmonic domination in Banach spaces

We show that if X is a Banach space with a Schauder basis, Ω ⊂ X is a pseudoconvex open subset, and u : Ω → ( − ∞ , ∞ ) is a locally bounded function, then there is a continuous plurisubharmonic function w : Ω → ( − ∞ , ∞ ) with u ( x ) ⩽ w ( x ) for all x ∈ Ω . This has many applications to analytic cohomology of complex Banach manifolds.

On complex Banach manifolds similar to Stein manifolds

We give an abstract definition, similar to the axioms of a Stein manifold, of a class of complex Banach manifolds in such a way that a manifold belongs to the class if and only if it is biholomorphic to a closed split complex Banach submanifold of a separable Banach space.

Extensions of holomorphic motions and holomorphic families of Möbius groups

A normalized holomorphic motion of a closed set in the Riemann sphere, defined over a simply connected complex Banach manifold, can be extended to a normalized quasiconformal motion of the sphere, in the sense of Sullivan and Thurston. In this paper, we show that if the given holomorphic motion, defined over a simply connected complex Banach manifold, has a group equivariance property, then the extended (normalized) quasiconformal motion will have the same property. We then deduce a generalization of a theorem of Bers on holomorphic families of isomorphisms of Mobius groups. We also obtain some new results on extensions of holomorphic motions. The intimate relationship between holomorphic motions and Teichmuller spaces is exploited throughout the paper.

Fiber structure and local coordinates for the Teichmüller space of a bordered Riemann surface

We show that the infinite-dimensional Teichmüller space of a Riemann surface whose boundary consists of n n closed curves is a holomorphic fiber space over the Teichmüller space of an n n -punctured surface. Each fiber is a complex Banach manifold modeled on a two-dimensional extension of the universal Teichmüller space. The local model of the fiber, together with the coordinates from internal Schiffer variation, provides new holomorphic local coordinates for the infinite-dimensional Teichmüller space.

A complex structure on the set of quasiconformally extendible non-overlapping mappings into a Riemann surface

Let Σ be a Riemann surface with n distinguished points p 1,..., p n . We prove that the set of n-tuples (φ1,..., φ n ) of univalent mappings φ i from the unit disc $$ \mathbb{D} $$ into Σ mapping 0 to p i , with non-overlapping images and quasiconformal extensions to a neighbourhood of $$ \overline {\mathbb{D}} $$ , carries a natural complex Banach manifold structure. This complex structure is locally modeled on the n-fold product of a two complex-dimensional extension of the universal Teichmuller space. Our results are motivated by Teichmuller theory and two-dimensional conformal field theory.

On normal and non-normal holomorphic functions on complex Banach manifolds

Let X be a complex Banach manifold. A holomorphic function f : X → C is called a normal function if the family F f = { f ◦φ : φ ∈ O( , X)} forms a normal family in the sense of Montel (here O( , X) denotes the set of all holomorphic maps from the complex unit disc into X ). Characterizations of normal functions are presented. A sufficient condition for the sum of a normal function and non-normal function to be non-normal is given. Criteria for a holomorphic function to be non-normal are obtained. These results are used to draw one interesting conclusion on the boundary behavior of normal holomorphic functions in a convex bounded domain D in a complex Banach space V . Let {xn} be a sequence of points in D which tends to a boundary point ξ ∈ ∂ D such that limn→∞ f (xn) = L for some L ∈ C. Sufficient conditions on a sequence {xn} of points in D and a normal holomorphic function f are given for f to have the admissible limit value L , thus extending the result obtained by Bagemihl and Seidel. Mathematics Subject Classification (2000): 32A18 (primary).

A converse to the Oka principle over certain complex Banach manifolds

First Kajiwara then Leiterer gave geometric or cohomological criteria in the spirit of the Grauert–Oka principle for an open subset D of a Stein manifold M to be itself Stein. We give here criteria analogous to Leiterer's, e.g., for a relatively open subset D of a closed complex Hilbert submanifold M of separable Hilbert space to be itself biholomorphic to a closed complex Hilbert submanifold of separable Hilbert space.