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Hideki Miyachi

Publications and source records attributed to Hideki Miyachi.

27 records · Page 2Linked to original sources
arXiv2019

Pluripotential theory on Teichmüller space I -- Pluricomplex Green function

This is the first paper in a series of investigation of the pluripotential theory on Teichmüller space. The main purpose of this paper is to give an alternative approach to the Krushkal formula of the pluricomplex Green function on Teichmüller space. We also show that Teichmüller space carries a natural stratified structure of real-analytic submanifolds defined from the structure of singularities of the initial differentials of the Teichmüller mappings from a given point. We will also give a description of the Levi form of the pluricomplex Green function using the Thurston symplectic form via the Dumas symplectic structure on the space of holomorphic quadratic differentials.

math.CV↗
arXiv2019

Convergence of Teichmüller deformations in the universal Teichmüller space

Let $φ:\mathbb{D}\to\mathbb{C}$ be an integrable holomorphic function on the unit disk $\mathbb{D}$ and $D_φ:\mathbb{D}\to T(\mathbb{D})$ the Teichmüller disk in the universal Teichmüller space $T(\mathbb{D})$. For a positive $t$ it is known that $D_φ(t)\to [μ_φ]\in PML_b(\mathbb{D})$ as $t\to 1$, where $μ_φ$ is a bounded measured lamination representing a point on the Thurston boundary of $T(\mathbb{D})$. We extend this result by showing that $D_φ\colon \mathbb{D}\to T(\mathbb{D})$ extends as a continuous map from the closed disk $\overline{\mathbb{D}}$ to the Thurston bordification. In addition, we prove that the rate of convergence of $D_φ(λ)$ when $λ\to e^{iθ}$ is independent of the type of the approach to $e^{iθ}\in\partial\mathbb{D}$.

math.GT↗
arXiv2015

Geometry of the Gromov product : Geometry at infinity of Teichmüller space

This paper is devoted to study of transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We characterize a quotient semigroup of such transformations on Teichmüller space by use of simplicial automorphisms of the complex of curves, and we will see that such transformation is recognized as a "coarsification" of isometries on Teichmüller space which is rigid at infinity. We also show a hyperbolic characteristic that any finite dimensional Teichmüller space does not admit (quasi)-invertible rough-homothety.

math.MG↗
arXiv2015

Extremal length functions are log-plurisubharmonic

In this paper, we show that the extremal length functions on Teichmüller space are log-plurisubharmonic. As a corollary, we obtain an alternative proof of L.Liu and W.Su's results on the plurisubharmonicity of extremal length functions. We also obtain alternative proofs of S.Krushkal's results that a function defined by the Teichmüller distance from a reference point is plurisubharmonic, and the Teichmüller space is hyperconvex. To show the log-plurisubharmonicity, we give an explicit formula of the Levi form of the extremal length functions in generic case.

math.CV↗
arXiv2014

A rigidity theorem for holomorphic disks in Teichmüller space

In this paper, we discuss a rigidity property for holomorphic disks in Teichmüller space. In fact, we give a refinement of Tanigawa's rigidity theorem. We will also treat the rigidity property of holomorphic disks for complex manifolds. We observe the rigidity property is valid for bounded strictly pseudoconvex domains with $C^{2}$-boundaries, but the rigidity property does not hold for product manifolds.

math.CV↗
arXiv2014

Unification of extremal length geometry on Teichmuller space via intersection number

In this paper, we give a framework for the study of the extremal length geometry of Teichmüller space after S. Kerckhoff, F. Gardiner and H. Masur. There is a natural compactification using extremal length geometry introduced by Gardiner and Masur. The compactification is realized in a certain projective space. We develop the extremal length geometry in the cone which is defined as the inverse image of the compactification via the quotient mapping. The compactification is identified with a subset of the cone by taking an appropriate lift. The cone contains canonically the space of measured foliations in the boundary. We first extend the geometric intersection number on the space of measured foliations to the cone, and observe that the restriction of the intersection number to Teichmüller space is represented explicitly by the formula in terms of the Gromov product with respect to the Teichmüller distance. From this observation, we deduce that the Gromov product extends continuously to the compactification. As an application, we obtain an alternative approach to Earle-Ivanov-Kra-Markovic-Royden's characterization of isometries. Namely, with some few exceptions, the isometry group of Teichmüller space with respect to the Teichmüller distance is canonically isomorphic to the extended mapping class group. We also obtain a new realization of Teichmüller space, a hyperboloid model of Teichmüller space with respect to the Teichmüller distance.

math.GT↗
arXiv2010

Gardiner-Masur boundary of Teichmuller space : Vanishing subsurfaces and Uniquely ergodic boundary points

In this paper, we investigate the structure of the Gardiner-Masur boundary of Teichmuller space. Indeed, we will give a geometric description of boundary comparing to the Duchin-Leininger-Rafi compactification of the space of singular flat structures. We will obtain the coincidence between the Gardiner-Masur boundary and the Thurston boundary at the projective classes of uniquely ergodic measured foliations. We also study the action of the mapping class group on the Gardiner-Masur boundary and characterize the elements by fixed points. This paper has been withdrawn by the author. This paper will be renewal and the author will publish extended results elsewhere. If someone wants to see this preprint, please ask and consult the author.

math.GT↗
arXiv2010

A Fréchet topology on measured laminations and Earthquakes in the hyperbolic plane

We prove that the bijective correspondence between the space of bounded measured laminations $ML_b(\mathbb{H})$ and the universal Teichmüller space $T(\mathbb{H})$ given by $λ\mapsto E^λ|_{S^1}$ is a homeomorphism for the Fréchet topology on $ML_b(\mathbb{H})$ and the Teichmüller topology on $T(\mathbb{H})$, where $E^λ$ is an earthquake with earthquake measure $λ$. A corollary is that earthquakes with discrete earthquake measures are dense in $T(\mathbb{H})$. We also establish infinitesimal versions of the above results.

math.GT↗