arXiv subjects

Katsuhiko Matsuzaki

Publications and source records attributed to Katsuhiko Matsuzaki.

At least 37 records · Page 2Linked to original sources

arXiv2022

Parametrization of the $p$-Weil-Petersson curves: holomorphic dependence

Similarly to the Bers simultaneous uniformization, the product of the $p$-Weil-Petersson Teichmüller spaces for $p \geq 1$ provides the coordinates for the space of $p$-Weil-Petersson embeddings $γ$ of the real line $\mathbb R$ into the complex plane $\mathbb C$. We prove the biholomorphic correspondence from this space to the $p$-Besov space of $u=\log γ'$ on $\mathbb R$ for $p>1$. From this fundamental result, several consequences follow immediately which clarify the analytic structures concerning parameter spaces of $p$-Weil-Petersson curves. In particular, it follows that the correspondence of the Riemann mapping parameters to the arc-length parameters keeping the images of curves is a homeomorphism with bi-real-analytic dependence of change of parameters. This is a counterpart to a classical theorem of Coifman and Meyer for chord-arc curves.

math.CV↗

arXiv2022

Strongly symmetric homeomorphisms on the real line with uniform continuity

We investigate strongly symmetric homeomorphisms of the real line which appear in harmonic analysis aspects of quasiconformal Teichmüller theory. An element in this class can be characterized by a property that it can be extended quasiconformally to the upper half-plane so that its complex dilatation induces a vanishing Carleson measure. However, differently from the case on the unit circle, strongly symmetric homeomorphisms on the real line are not preserved under either the composition or the inversion. In this paper, we present the difference and the relation between these two cases. In particular, we show that if uniform continuity is assumed for strongly symmetric homeomorphisms of the real line, then they are preserved by those operations. We also show that the barycentric extension of uniformly continuous one induces a vanishing Carleson measure and so do the composition and the inverse of those quasiconformal homeomorphisms of the upper half-plane.

math.CV↗

arXiv2022

Chordal Loewner chains and Teichmüller spaces on the half-plane

We consider a univalent analytic function $f$ on the half-plane satisfying the condition that the supremum norm of its (pre-)Schwarzian derivative vanishes on the boundary. Under certain extra assumptions on $f$, we show that there exists a chordal Loewner chain initiated from $f$ until some finite time, and this Loewner chain defines a quasiconformal extension of $f$ over the boundary such that its complex dilatation is given explicitly in terms of the (pre-)Schwarzian derivative in some neighborhood of the boundary. This can be regarded as the half-plane version of the corresponding result developed on the disk by Becker and also the generalization of the Ahlfors-Weill formula. As an application of this quasiconformal extension, we complete the characterization of an element of the VMO-Teichmüller space on the half-plane using the vanishing Carleson measure condition induced by the (pre-)Schwarzian derivative.

math.CV↗

arXiv2022

The $p$-Weil-Petersson Teichmüller space and the quasiconformal extension of curves

We consider the correspondence between the space of $p$-Weil-Petersson curves $γ$ on the plane and the $p$-Besov space of $u=\log γ'$ on the real line for $p >1$. We prove that the variant of the Beurling-Ahlfors extension defined by using the heat kernel yields a holomorphic map for $u$ on a domain of the $p$-Besov space to the space of $p$-integrable Beltrami coefficients. This in particular gives a global real-analytic section for the Teichmüller projection from the space of $p$-integrable Beltrami coefficients to the $p$-Weil-Petersson Teichmüller space.

math.CV↗

arXiv2021

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, Kenig and Pipher in 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.

math.CV↗

arXiv2021

BMO embeddings, chord-arc curves, and Riemann mapping parametrization

We consider the space of chord-arc curves on the plane passing through the infinity with their parametrization $γ$ on the real line, and embed this space into the product of the BMO Teichmüller spaces. The fundamental theorem we prove about this representation is that $\log γ'$ also gives a biholomorphic homeomorphism into the complex Banach space of BMO functions. Using these two equivalent complex structures, we develop a clear exposition on the analytic dependence of involved mappings between certain subspaces. Especially, we examine the parametrization of a chord-arc curve by using the Riemann mapping and its dependence on the arc-length parametrization. As a consequence, we can solve a conjecture of Katznelson, Nag, and Sullivan in 1990 by showing that this dependence is not continuous.

math.CV↗

arXiv2020

Beurling-Ahlfors extension by heat kernel, ${\rm A}_\infty$-weights for VMO, and vanishing Carleson measures

We investigate a variant of the Beurling-Ahlfors extension of quasisymmetric homeomorphisms of the real line that is given by the convolution of the heat kernel, and prove that the complex dilatation of such a quasiconformal extension of a strongly symmetric homeomorphism (i.e. its derivative is an ${\rm A}_\infty$-weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half-plane.

math.CV↗

arXiv2020

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 $α$-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 $C^{1+α}$-topology at the base point.

math.CV↗

arXiv2020

Rigidity of groups of circle diffeomorphisms and Teichmüller spaces

We consider deformations of a group of circle diffeomorphisms with Hölder continuous derivatives in the framework of quasiconformal Teichmüller theory and show certain rigidity under conjugation by symmetric homeomorphisms of the circle. As an application, we give a condition for such a diffeomorphism group to be conjugate to a Möbius group by a diffeomorphism of the same regularity. The strategy is to find a fixed point of the group which acts isometrically on the integrable Teichmüller space with the Weil-Petersson metric.

math.CV↗

arXiv2019

Normalizer, divergence type and Patterson measure for discrete groups of the Gromov hyperbolic space

For a non-elementary discrete isometry group $G$ of divergence type acting on a proper geodesic $δ$-hyperbolic space, we prove that its Patterson measure is quasi-invariant under the normalizer of $G$. As applications of this result, we have: (1) under a minor assumption, such a discrete group $G$ admits no proper conjugation, that is, if the conjugate of $G$ is contained in $G$, then it coincides with $G$; (2) the critical exponent of any non-elementary normal subgroup of $G$ is strictly greater than half of that for $G$.

math.MG↗

arXiv2019

Symmetric and strongly symmetric homeomorphisms on the real line with non-symmetric inversion

We show an example of a symmetric homeomorphism $h$ of the real line $\mathbb{R}$ onto itself such that $h^{-1}$ is not symmetric. This implies that the set of all symmetric self-homeomorphisms of $\mathbb{R}$ does not constitute a group under the composition. We also deal with strongly symmetric self-homeomorphisms of $\mathbb{R}$ along the same line. These results reveal the difference of the sets of such self-homeomorphisms of the real line from those of the unit circle.

math.CV↗

arXiv2019

Teichmüller spaces of piecewise symmetric homeomorphisms on the unit circle

We interpolate a new family of Teichmüller spaces $T_{\sharp}^X$ between the universal Teichmüller space $T$ and its little subspace $T_0$, which we call the Teichmüller space of piecewise symmetric homeomorphisms. This is defined by prescribing a subset $X$ of the unit circle. The inclusion relation of $X$ induces a natural inclusion of $T_{\sharp}^X$, and an approximation of $T$ is given by an increasing sequence of $T_{\sharp}^X$. In this paper, we discuss the fundamental properties of $T_{\sharp}^X$ from the viewpoint of the quasiconformal theory of Teichmüller spaces. We also consider the quotient space of $T$ by $T_{\sharp}^X$ as an analog of the asymptotic Teichmüller space.

math.CV↗

arXiv2019

Teichmüller spaces of generalized symmetric homeomorphisms

We introduce the concept of a new kind of symmetric homeomorphisms 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 endow a complex Banach manifold structure on the space of those generalized symmetric homeomorphisms.

math.CV↗

arXiv2019

Injectivity of the quotient Bers embedding of Teichmüller spaces

The Bers embedding of theTeichmüller space is a homeomorphism into the Banach space of certain holomorphic automorphic forms. For a subspace of the universal Teichmüller space and its corresponding Banach subspace, we consider whether the Bers embedding can project down between their quotient spaces. If this is the case, it is called the quotient Bers embedding. Injectivity of the quotient Bers embedding is the main problem in this paper. Alternatively, we can describe this situation as the universal Teichmüller space having an affine foliated structure induced by this subspace. We give several examples of subspaces for which the injectivity holds true, including the Teichmüller space of circle diffeomorphisms with Hölder continuous derivative. As an application, the regularity of conjugation between representations of a Fuchsian group into the group of circle diffeomorphisms is investigated.

math.CV↗

arXiv2018

BMO Teichmüller spaces and their quotients with complex and metric structures

The paper presents some recent results on the BMO Teichmüller space, its subspaces and quotient spaces. We first consider the chord-arc curve subspace and prove that every element of the BMO Teichmüller space is represented by its finite composition. Moreover, we show that these BMO Teichmüller spaces have affine foliated structures induced by the VMO Teichmüller space. By which, their quotient spaces have natural complex structures modeled on the quotient Banach space. Then, a complete translation-invariant metric is introduced on the BMO Teichmüller space and is shown to be a continuous Finsler metric in a special case.

math.CV↗

arXiv2018

On horospheric limit sets of Kleinian groups

In this paper we partially answer a question of P. Tukia about the size of the difference between the big horospheric limit set and the horospheric limit set of a Kleinian group. We mainly investigate the case of normal subgroups of Kleinian groups of divergence type and show that this difference is of zero conformal measure by using another result obtained here: the Myrberg limit set of a non-elementary Kleinian group is contained in the horospheric limit set of any non-trivial normal subgroup.

math.CV↗

arXiv2018

Weighted cogrowth formula for free groups

We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group ${\rm Cay}(F_n)$ endowed with variable edge lengths, by an arbitrary subgroup $G$ of $F_n$. Our main result, which generalizes Grigorchuk's cogrowth formula to variable edge lengths, provides a formula relating the bottom of the spectrum of weighted Laplacian on $G \backslash {\rm Cay}(F_n)$ to the Poincaré exponent of $G$. Our main tool is the Patterson-Sullivan theory for Cayley graphs with variable edge lengths.

math.DS↗

arXiv2016

Continuity of the barycentric extension of circle diffeomorphisms of Hölder continuous derivatives

The barycentric extension due to Douady and Earle gives a conformally natural extension of a quasisymmetric automorphism of the circle to a quasiconformal automorphism of the unit disk. We consider such extensions for circle diffeomorphisms of Hölder continuous derivatives and show that this operation is continuous with respect to an appropriate topology for the space of the corresponding Beltrami coefficients.

math.CV↗