scholarly journals VMO-Teichmüller space on the real line

2021 ◽  
Vol 47 (1) ◽  
pp. 57-82
Author(s):  
Yuliang Shen
Keyword(s):  
Real Line ◽  
Carleson Measure ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Banach Manifold ◽  
Quasiconformal Homeomorphism ◽  
The Real ◽  
Beltrami Coefficient ◽  
Upper Half Plane ◽  
Quasisymmetric Homeomorphism

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.  

scholarly journals On boundaries of Schottky spaces

1976 ◽  
Vol 62 ◽  
pp. 97-124 ◽  
Author(s):  
Hiroki Sato
Keyword(s):  
Riemann Surface ◽  
Half Plane ◽  
Fuchsian Group ◽  
Compact Riemann Surface ◽  
Teichmüller Space ◽  
Schottky Group ◽  
Teichmuller Space ◽  
Upper Half Plane

Let S be a compact Riemann surface and let Sn be the surface obtained from S in the course of a pinching deformation. We denote by Γn the quasi-Fuchsian group representing Sn in the Teichmüller space T(Γ), where Γ is a Fuchsian group with U/Γ = S (U: the upper half plane). Then in the previous paper [7] we showed that the limit of the sequence of Γn is a cusp on the boundary ∂T(Γ). In this paper we will consider the case of Schottky space . Let Gn be a Schottky group with Ω(Gn)/Gn = Sn. Then the purpose of this paper is to show what the limit of Gn is.

scholarly journals Existence of extremal Beltrami coefficients with nonconstant modulus

2010 ◽  
Vol 199 ◽  
pp. 1-14 ◽  
Author(s):  
Guowu Yao
Keyword(s):  
Open Problem ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Constant Modulus ◽  
Beltrami Coefficient ◽  
Extremal Sets ◽  
Universal Teichmüller Space

AbstractSuppose that [μ]T(Δ) is a point of the universal Teichmüller space T(Δ). In 1998, Božin, Lakic, Marković, and Mateljević showed that there exists μ such that μ is uniquely extremal in [μ]T(Δ) and has a nonconstant modulus. It is a natural problem whether there is always an extremal Beltrami coefficient of constant modulus in [μ]T(Δ) if [μ]T(Δ) admits infinitely many extremal Beltrami coefficients; the purpose of this paper is to show that the answer is negative. An infinitesimal version is also obtained. Extremal sets of extremal Beltrami coefficients are considered, and an open problem is proposed. The key tool of our argument is Reich’s construction theorem.

A Hilbert manifold structure on the Weil–Petersson class Teichmüller space of bordered Riemann surfaces

2015 ◽  
Vol 17 (04) ◽  
pp. 1550016 ◽  
Author(s):  
David Radnell ◽  
Eric Schippers ◽  
Wolfgang Staubach
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Banach Manifold ◽  
Holomorphic Maps ◽  
Discontinuous Group ◽  
Hilbert Manifold ◽  
Conformal Field ◽  
Compact Riemann Surfaces

We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disk removed. We define a refined Teichmüller space of such Riemann surfaces (which we refer to as the WP-class Teichmüller space) and demonstrate that in the case that 2g + 2 - n > 0, this refined Teichmüller space is a Hilbert manifold. The inclusion map from the refined Teichmüller space into the usual Teichmüller space (which is a Banach manifold) is holomorphic. We also show that the rigged moduli space of Riemann surfaces with non-overlapping holomorphic maps, appearing in conformal field theory, is a complex Hilbert manifold. This result requires an analytic reformulation of the moduli space, by enlarging the set of non-overlapping mappings to a class of maps intermediate between analytically extendible maps and quasiconformally extendible maps. Finally, we show that the rigged moduli space is the quotient of the refined Teichmüller space by a properly discontinuous group of biholomorphisms.

scholarly journals Existence of extremal Beltrami coefficients with nonconstant modulus

2010 ◽  
Vol 199 ◽  
pp. 1-14 ◽  
Author(s):  
Guowu Yao
Keyword(s):  
Open Problem ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Constant Modulus ◽  
Beltrami Coefficient ◽  
Extremal Sets ◽  
Universal Teichmüller Space

AbstractSuppose that [μ]T(Δ)is a point of the universal Teichmüller spaceT(Δ). In 1998, Božin, Lakic, Marković, and Mateljević showed that there existsμsuch thatμis uniquely extremal in [μ]T(Δ)and has a nonconstant modulus. It is a natural problem whether there is always an extremal Beltrami coefficient of constant modulus in [μ]T(Δ)if [μ]T(Δ)admits infinitely many extremal Beltrami coefficients; the purpose of this paper is to show that the answer is negative. An infinitesimal version is also obtained. Extremal sets of extremal Beltrami coefficients are considered, and an open problem is proposed. The key tool of our argument is Reich’s construction theorem.

scholarly journals On monodromy map

1993 ◽  
Vol 16 (4) ◽  
pp. 695-708
Author(s):  
Jharna D. Sengupta
Keyword(s):  
Vector Bundle ◽  
Conjugacy Class ◽  
Equivalence Class ◽  
Half Plane ◽  
Fuchsian Group ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Projective Structures ◽  
Upper Half Plane ◽  
Monodromy Map

LetΓbe a Fuchsian group acting on the upper half-planeUand having signature{p,n,0;v1,v2,…,vn};2p−2+∑j=1n(1−1vj)>0.LetT(Γ)be the Teichmüller space ofΓ. Then there exists a vector bundleℬ(T(Γ))of rank3p−3+noverT(Γ)whose fibre over a pointt∈T(Γ)representingΓtis the space of bounded quratic differentialsB2(Γt)forΓt. LetHom(Γ,G)be the set of all homomorphisms fromΓinto the Mbius groupG.For a given(t,ϕ)∈ℬ(T(Γ))we get an equivalence class of projective structures and a conjugacy class of a homomorphismx∈Hom(Γ,G). Therefore there is a well defined mapΦ:ℬ(T(Γ))→Hom(Γ,G)/G,Φis called the monodromy map. We prove that the monromy map is hommorphism. The casen=0gives the previously known result by Earle, Hejhal Hubbard.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

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