Hideki Miyachi

On invariant distances on asymptotic Teichmuller spaces

In this paper, we will establish that any invariant distance on asymptotic Teichmuller space is a complete distance.

Cusps in complex boundaries of one-dimensional Teichmüller space

This paper gives a concrete proof of the conjectural phenomena on the complex boundary one-dimensional slices: every rational boundary point is cusp shaped. This paper treats this problem for Bers slices, the Earle slice, and the Maskit slice. In proving this, we also show that every Teichmüller modular transformation acting on a Bers slice can be extended as a quasiconformal mapping on its ambient space. Furthermore, using this extension, we discuss similarity phenomena on the boundaries of Bers slices, and also compare these phenomena with results in complex dynamics. We will also give a result, related to the theory of L. Keen and C. Series, of pleated varieties in quasifuchsian space of once punctured tori.

On the horocyclic coordinate for the Teichmüller space of once punctured tori

This paper deals with analytic and geometric properties of the Maskit embedding of the Teichmuller space of once punctured tori. We show that the image of this embedding has an inward-pointing cusp and study the boundary behavior of conformal automorphisms. These results are proved using Y.N. Minskys Pivot Theorem.

The inner and outer radii of asymptotic Teichmüller spaces

In this article, we will show that for a Riemann surface R, the inner and outer radii of the asymptotic Teichmüller space AT(R) of R are extremal when R satisfies one of Nakanishi–Yamamotos conditions (O1) and (O2), and vice versa. This kind of characterization is first observed by Nakanishi and Yamamoto for the outer radii of Teichmüller spaces. We will also observe that the inner and outer radii of AT(R) are attained at a point in the boundary of AT(R) when R satisfies either (O1) or (O2).

Extremal length boundary of the Teichmüller space contains non-Busemann points

We present an overview of the extremal length embedding of a Teichmüller space and its extremal length compactification. For Teichmüller spaces of dimension at least two, we describe a large class of non-Busemann points on the metric boundary, that is, points that cannot be realized as limits of almost geodesic rays.

Holonomies and the slope inequality of Lefschetz fibrations

In this paper, we consider two necessary conditions: the irreducibility of the holonomy group and the slope inequality for which a Lefschetz fibration over a closed orientable surface is a holomorphic fibration. We show that these two conditions are independent in the sense that neither one implies the other.

On extendibility of a map induced by the Bers isomorphism

Let T (S) be the Teichmuller space of a closed Riemann surface S of genus g(> 1). Denote by U the universal covering of S, that is, the upper halfplane and denote by Ṡ the surface obtained by removing a point from S. By Bers isomorphism theorem, we have a map from T (S)×U to T (Ṡ). It is known that the Teichmuller space T (Ṡ) is embedded in (3g− 2)-dimensional complex vector space. Thus the boundary ∂T (Ṡ) of T (Ṡ) is naturally defined. Let A be a subset of ∂U consisting of all points filling S. In this talk, we show that the map of T (S) × U to T (Ṡ) has a continuous extension of T (S) × (U ∪ A) into T (Ṡ) ∪ ∂T (Ṡ). This is a joint work with Hideki Miyachi (Osaka University).

A characterization of biholomorphic automorphisms of Teichmüller space

In this paper, we give an alternative approach to Royden–Earle–Kra–Markovics characterization of biholomorphic automorphisms of Teichmuller space of Riemann surface of analytically finite type.