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Hideki Miyachi

Publications and source records attributed to Hideki Miyachi.

At least 19 recordsLinked to original sources
arXiv2026

Hopf decomposition of the actions of subgroups of the mapping class group

We study the Hopf decomposition of subgroup actions of the Teichmüller modular group on the Thurston boundary with respect to the Thurston measure class. Kaimanovich's general Radon--Nikodym criteria allow us to describe the conservative and dissipative parts by the divergence and convergence of a series expressed in terms of extremal length. We identify the conservative part with the big horospherical limit set modulo null sets. For any basepoint with trivial stabilizer in the subgroup, we identify the dissipative part, modulo null sets, with the set of Dirichlet points and with the union of the subgroup translates of the ideal boundary of the associated Dirichlet polyhedron. The description via Dirichlet polyhedra uses the fact that level sets of extremal-length ratios at distinct points of Teichmüller space have measure zero. For the Torelli group of a closed surface of genus at least two, we use radial limits of the period map to prove that its conical limit set has measure zero. Combining our geometric characterization with the conservativity of its boundary action established by Choi, Gekhtman, Yang, and Zheng, we obtain that its big horospherical limit set has full measure and that the ideal boundary of each Dirichlet polyhedron has measure zero.

math.GT↗
arXiv2026

Limit sets of mapping class groups

We introduce the notions of horospherical limit point and big horospherical limit point in the projective measured foliation space for a subgroup of the mapping class group, which are analogous to these notions for hyperbolic spaces. We classify Teichmüller geodesic rays directed by projective measured foliations into conical, horospherical, big horospherical limit points, and show that these can be determined by the topological properties of the directing foliations: except for the case of minimal and uniquely ergodic projective measured foliation, whose corresponding ray can be either a conical limit point or a non-conical horospherical limit point.

math.GT↗
arXiv2026

On the construction of geographical maps: Lagrange, Chebyshev, Darboux and Milnor

Lagrange, Chebyshev, and Darboux, in 1779, 1856, and 1911, respectively, wrote articles all bearing the same title, \emph{On the Construction of Geographical Maps}. In 1969, Milnor wrote a paper in which he refers to Chebyshev's paper, of which he provides a new formulation and proof. In the present article, we review the results of all these papers, explaining the main ideas they contain and pointing out connections between them. We give complete proofs of the statements by Darboux and Milnor, both of which aim to make explicit and provide a proof of Chebyshev's result, but whose contents are different. Although Chebyshev did not state explicitly what the word ``best'' means, his conclusion, like that of Darboux and of Milnor, is that a best geographical map is characterised by the fact that its conformal factor is constant on the boundary of the region represented. Our statement and proof of Milnor's theorem work in a more general setting than the one he gives. The final version of this paper will appear in the Handbook of Mathematics in the Arts and Sciences (second edition), ed. Bharath Sriraman, Springer, 2027.

math.DG↗
arXiv2026

The horocyclic metric on Teichm{ü}ller spaces

In his paper Minimal stretch maps between hyperbolic surfaces, William Thurston defined a norm on the tangent space to Teichm{ü}ller space of a hyperbolic surface, which he called the earthquake norm. This norm is obtained by assigning a length to a tangent vector after such a vector is considered as an infinitesimal earthquake deformation of the surface. This induces a Finsler metric on the Teichm{ü}ller space, called the earthquake metric. This theory was recently investigated by Huang, Ohshika, Pan and Papadopoulos. In the present paper, we study this metric from the conformal viewpoint and we adapt Thurston's theory to the case of Riemann surfaces of arbitrary genus with marked points. A complex version of the Legendre transform defined for Finsler manifolds gives an analogue of the Wolpert duality for the Weil-Petersson symplectic form, which establishes a complete analogue of Thurston's theory of the earthquake norm in the conformal setting. This paper will appear in the Annales de l'Institut Fourier

math.CV↗
arXiv2026

On the Lambert conformal conical projection and the general map of the Russian Empire

The problem of drawing geographical maps is the one of mapping a subset of the sphere, representing a country or some other region on the surface of the Earth, into the Euclidean plane, minimising certain distortion properties that are specified in advance. It is known that from the purely mathematical point of view, this is an extremely difficult problem. One of Leonhard Euler's duties during his first stay at the Imperial Academy of Sciences of Saint Petersburg (1727-1741) was to help establishing maps of the Russian Empire. He worked on this project under the direction of the famous French geographer Joseph-Nicolas Delisle, who was the head of the astronomy and geography departments of the Academy. The general map of the Russian Empire, together with several maps of its particular regions were published under Euler's direction in the so-called Russian Atlas in 1745. In his later memoir ``De proiectione geographica De Lisliana in mappa generali imperii russici usitata'', written in 1777, Euler developed the mathematical theory of the method used by Delisle on a heuristic basis, which he himself used for drawing the general map of the Russian Empire. This method usually carries now the name Delisle--Euler map. In a previous paper, the first two authors of the present paper compared the Delisle--Euler map with several other maps of the conical type, with respect to various mathematical distorsion properties. They showed that this map is the best one from all the points of view considered, when it is applied to the drawing of the Russian Empire. In the present paper, we compare the Euler--Delisle map with a map which was not considered in the paper mentioned, namely, the so-called Lambert conformal conical projection, applied to the same region of the Earth. We show that the latter is better in several respects than all the other maps considered in the previous paper, including the Delisle--Euler map.

math.DG↗
arXiv2025

The asymptoticity of pairs of Teichmüller rays

In this paper, we study the limit of Teichmüller distance between two points along a pair of Teichmüller rays. We obtain an explicit formula for the limiting Teichmüller distance when the vertical measured foliations of the quadratic differentials are finite sums of weighted simple closed curves and uniquely ergodic measures. The limit is expressed in terms of ratios of the corresponding moduli and the Teichmüller distance between the limit surfaces when the vertical measured foliations are absolutely continuous. Consequently, two Teichmüller rays are asymptotic if and only if their vertical measured foliations are modularly equivalent and their limit surfaces coincide, which implies a main result of Masur on the asymptoticity of Teichmüller rays determined by uniquely ergodic quadratic differentials. Furthermore, we prove that the infimum of the limiting Teichmüller distances can be represented in terms of the distance between the limit surfaces of the Teichmüller rays and the detour metric of their endpoints on the Gardiner-Masur boundary, when the initial points of the rays vary along the Teichmüller geodesics.

math.CV↗
arXiv2025

Function theory, Dynamics and Ergodic theory via Thurston theory

In this paper, we discuss function theory on Teichmüller space through Thurston's theory, as well as the dynamics of subgroups of the mapping class group of a surface, with reference to Sullivan's theory on the ergodic actions of discrete subgroups of the isometry group of hyperbolic space at infinity.

math.CV↗
arXiv2025

Benoist-Hulin groups

A Benoist-Hulin group is, by definition, a subgroup $Γ$ of ${\rm PSL}_2(\mathbb{C})$ such that any $Γ$-invariant closed set consisting of Jordan curves in the space of closed subsets of the Riemann sphere that are not singletons is composed of $K$-quasicircles for some $K \ge 1$. Y.Benoist and D.Hulin showed that the full group ${\rm PSL}_2(\mathbb{C})$ is a Benoist-Hulin group. In this paper, we develop the theory of Benoist-Hulin groups and show that both uniform lattices and parabolic subgroups are Benoist-Hulin groups.

math.GR↗
arXiv2025

Bounded Pluriharmonic Functions and Holomorphic Functions on Teichmüller Space II -- Poisson integral formula --

In this paper, we establish the Poisson integral formula for bounded pluriharmonic functions on the Teichmüller space of analytically finite Riemann surfaces of type $(g,m)$ with $2g-2+m>0$. We also discuss a version of the F. and M. Riesz theorem concerning the value distribution of plurisubharmonic functions on the Teichmüller space, as well as a Teichmüller-theoretic interpretation of the mean value theorem for pluriharmonic functions.

math.CV↗
arXiv2025

On the Marinus--Ptolemy and Delisle--Euler conical maps

We examine connections between the mathematics behind methods of drawing geographical maps due, on the one hand to Marinos and Ptolemy (1st-2nd c. CE) and on the other hand to Delisle and Euler (18th century). A recent work by the first two authors of this article shows that methods of Delisle and Euler for drawing geographical maps, which are improvements of methods of Marinus and Ptolemy, are best among a collection of geographical maps we term ``conical''. This is an instance where after practitioners and craftsmen (here, geographers) have used a certain tool during several centuries, mathematicians prove that this tool is indeed optimal. Many connections among geography, astronomy and geometry are highlighted. The fact that the Marinos--Ptolemy and the Delisle--Euler methods of drawing geographical maps share many non-trivial properties is an important instance of historical continuity in mathematics.

math.HO↗
arXiv2024

Bounded pluriharmonic functions and holomorphic functions on Teichmüller space

In this paper, we discuss the boundary behavior of bounded pluriharmonic functions on the Teichmüller space. We will show a version of the Fatou theorem that every bounded pluriharmonic function admits the radial limits along the Teichmüller geodesic rays, and a version of the F. and M. Riesz theorem that the radial limit of a non-constant bounded holomorphic function is not constant on any non-null measurable set on the Bers boundary in terms of the pluriharmonic measure. As a corollary, we obtain the non-ergodicity of the action of the Torelli group for a closed surface of genus $g\ge 2$ on the space of projective measured foliations.

math.CV↗
arXiv2024

The $L^1$-$L^\infty$-geometry of Teichmüller space -- Second order infinitesimal structures

The $L^1$-$L^\infty$ geometry is the Finsler geometry of the Teichmüller space by the Teichmüller metric and the $L^1$-norm function of holomorphic quadratic differentials. In this paper, aiming to develop the $L^1$-$L^\infty$-geometry and the differential geometry on the Teichmüller space, we formulate the second order infinitesimal structures (the infinitesimal structures on the (co)tangent bundles) over the Teichmüller space. We will give model spaces of the second order infinitesimal spaces. By applying our formulation, we give affirmative answers to two folklore. We first show that the map from the space of holomorphic quadratic differentials to the tangent bundle defined by Teichmüller Beltrami differentials is a real-analytic diffeomorphism on every stratum in the space of holomorphic quadratic differentials. Second, we show that the Teichmüller metric is real-analytic on the image of each stratum. We also observe a new duality between the Teichmüller metric and the $L^1$-norm function at the infinitesimal level.

math.CV↗
arXiv2023

On the Teichm{ü}ller space of acute triangles

We continue the study of the analogue of Thurston's metric on the Teichm{ü}ller space of Euclidean triangle which was started by Saglam and Papadopoulos in [1].By direct calculation, we give explicit expressions of the distance function and the Finsler structure of the metric restricted to the subspace of acute triangles.We deduce from the form of the Finsler unit sphere a result on the infinitesimal rigidity of the metric.We give a description of the maximal stretching loci for a family of extreme Lipschitz maps.

math.GT↗
arXiv2022

Pluripotential theory on Teichmüller space II -- Poisson integral formula

This is the second paper in a series of investigations of the pluripotential theory on Teichmüller space. The main purpose of this paper is to establish the Poisson integral formula for pluriharmonic functions on Teichmüller space which are continuous on the Bers compactification. We also observe that the Schwarz type theorem on the boundary behavior of the Poisson integral. We will see a relationship between the pluriharmonic measures and the Patterson-Sullivan measures discussed by Athreya, Bufetov, Eskin and Mirzakhani.

math.CV↗
arXiv2022

The Teichm{ü}ller-Randers metric

In this paper, we introduce a new asymmetric weak metric on the Teichm{ü}ller space of a closed orientable surface with (possibly empty) punctures.This new metric, which we call the Teichm{ü}ller-Randers metric, is an asymmetric deformation of the Teichm{ü}ller metric, and is obtained by adding to the infinitesimal form of the Teichm{ü}ller metric a differential 1-form. We study basic properties of the Teichm{ü}ller-Randers metric. In the case when the 1-form is exact, any Teichm{ü}ller geodesic between two points is a unique Teichm{ü}ller--Randers geodesic between them. A particularly interesting case is when the differential 1-form is (up to a factor) the differential of the logarithm of the extremal length function associated with a measured foliation. We show that in this case the Teichm{ü}ller-Randers metric is incomplete in any Teichm{ü}ller disc, and we give a characterisation of geodesic rays with bounded length in this disc in terms of their directing measured foliations.

math.CV↗
arXiv2021

Tangent spaces of the Teichm{ü}ller space of the torus with Thurston's weak metric

In this paper, we show that the analogue of Thurston's asymmetric metric on the Teichm{ü}ller space of flat structures on the torus is weak Finsler and we give a geometric description of its unit sphere at each point in the tangent space to Teichm{ü}ller space. We then introduce a family of weak Finsler metrics which interpolate between Thurston's asymmetric metric and the Teichm{ü}ller metric of the torus (which coincides with the the hyperbolic metric). We describe the infinitesimal unit spheres of the metrics in this family.The final version of this paper will appear in Annales Academiæ \ Scientiarum Fennicæ\ Mathematica.

math.CV↗
arXiv2020

Universal commensurability augmented Teichmüller space and moduli space

It is known that every finitely unbranched covering $α:\widetilde{S}_{g(α)}\rightarrow S$ of a compact Riemann surface $S$ with genus $g\geq2$ induces an isometric embedding $Γ_α$ from the Teichmüller space $T(S)$ to the Teichüller space $T(\widetilde{S}_{g(α)})$. Actually, it has been showed that the isometric embedding $Γ_α$ can be extended isometrically to the augmented Teichmüller space $\widehat{T}(S)$ of $T(S)$. Using this result, we construct a directed limit $\widehat{T}_{\infty}(S)$ of augmented Teichmüller spaces, where the index runs over all finitely unbranched coverings of $S$. Then, we show that the action of the universal commensurability modular group $Mod_{\infty}(S)$ can extend isometrically on $\widehat{T}_{\infty}(S)$. Furthermore, for any $X_{\infty}\in T_{\infty}(S)$, its orbit of the action of the universal commensurability modular group $Mod_{\infty}(S)$ on the universal commensurability augmented Teichmüller space $\widehat{T}_{\infty}(S)$ is dense. Finally, we also construct a directed limit $\widehat{M}_{\infty}(S)$ of augmented moduli spaces by characteristic towers and show that the subgroup $Caut(π_{1}(S))$ of $Mod_{\infty}(S)$ acts on $\widehat{T}_{\infty}(S)$ to produce $\widehat{M}_{\infty}(S)$ as the quotient.

math.GT↗