Hiroshige Shiga

A finiteness theorem for holomorphic families of Riemann surfaces

In this paper we will give an analytic proof of the following finiteness theorem for holomorphic families of Riemann surfaces: FIniteness Theorem OF Families. Let B be a Riemann surface of finite type. Then, there are only finitely many non-isomorphic and locally non-trivial holomorphic families of Riemann surfaces of fixed finite type (g, n) with 2g — 2 + n > 0 over B.

Extensions of holomorphic motions and holomorphic families of Möbius groups

A normalized holomorphic motion of a closed set in the Riemann sphere, defined over a simply connected complex Banach manifold, can be extended to a normalized quasiconformal motion of the sphere, in the sense of Sullivan and Thurston. In this paper, we show that if the given holomorphic motion, defined over a simply connected complex Banach manifold, has a group equivariance property, then the extended (normalized) quasiconformal motion will have the same property. We then deduce a generalization of a theorem of Bers on holomorphic families of isomorphisms of Mobius groups. We also obtain some new results on extensions of holomorphic motions. The intimate relationship between holomorphic motions and Teichmuller spaces is exploited throughout the paper. 1. Definitions and statements of the main theorems In their study of the dynamics of rational maps, Mane, Sad, and Sullivan introduced the idea of holomorphic motions (see [20]). Since then, holomorphic motions have found several interesting applications in Teichmuller theory, complex dynamics, and Kleinian groups. A central topic in the study of holomorphic motions is the question of extensions. In this paper, we obtain some new extension theorems. We also prove a generalization of a theorem of Bers on holomorphic families of isomorphisms of Mobius groups. 1.1. Holomorphic motions. DEFINITION 1.1. Let V be a connected complex manifold, and let E be a subset of O C. A holomorphic family of injections of E over V is a family of maps f x gx2V that has the following properties: (i) for each x in V , the map x W E ! O C is an injection, and, (ii) for each z in E , the map x 7! x (z) is holomorphic. 2000 Mathematics Subject Classification. Primary 32G15; Secondary 37F30, 37F45. The research of the first author was partially supported a PSC-CUNY Research Grant. He also wants to thank the very kind hospitality of the Department of Mathematics of Tokyo Institute of Technology, where most of this research was done. The research of the second author was supported by the Ministry of Education, Science, Sports and Culture, Japan, Grant-in-Aid for Scientific Research (A), 2005–2009, 17204010. 1168 S. MITRA AND H. SHIGA It is convenient to define W V E ! O C as the map (x , z) WD x (z) for all (x , z) 2 V E . If V is a connected complex manifold with a basepoint x0, then a holomorphic motion of E over V is a holomorphic family of injections such that (x0, z) D z for all z in E . A holomorphic motion W V E ! O C is called trivial if (x , z) D z for all (x , z) 2 V E . We say that V is the parameter space of the holomorphic motion . Unless otherwise stated, we will always assume that is a normalized holomorphic motion; i.e. 0, 1, and 1 belong to E and are fixed points of the map x ( ) for every x in V . DEFINITION 1.2. Let V and W be connected complex manifolds with basepoints, and f be a basepoint preserving holomorphic map of W into V . If W V E ! O C is a holomorphic motion, its pullback by f is the holomorphic motion f ( )(x , z) D ( f (x), z) for all (x , z) 2 W E of E over W . If E is a proper subset of Q E and W V E ! O C and Q W V Q E ! O C are two maps, we say that Q extends if Q (x , z) D (x , z) for all (x , z) in V E . If W V E ! O C is a holomorphic motion, it is natural to ask whether there exists a holomorphic motion Q W V O C! O C such that Q extends . For holomorphic motions over the open unit disk, the papers [5], [12], [20], [26], and [28] contain important results. Extensions of holomorphic motions over more general parameter spaces have been studied in the papers [13], [21], [22], and [23]. 1.2. Quasiconformal motions. In their paper [28], Sullivan and Thurston introduced the idea of quasiconformal motions. In what follows, denotes the Poincare metric on O C n f0, 1, 1g. Let V be a connected Hausdorff space with a basepoint x0. For any map W V E ! O C, x in V , and any quadruplet a, b, c, d of points in E , let x (a, b, c, d) denote the cross-ratio of the values (x , a), (x , b), (x , c), and (x , d). We will write (x , z) as x (z) for x in V and z in E . So we have: x (a, b, c, d) D ( x (a) x (c))( x (b) x (d)) ( x (a) x (d))( x (b) x (c)) (1.1)

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.

Holomorphic families of Riemann surfaces and monodromy

One fundamental theorem in the theory of holomorphic dynamics is Thurstons topological characterization of postcritically finite rational maps. Its proof is a beautiful application of Teichmuller theory. In this chapter we provide a self-contained proof of a slightly generalized version of Thurstons theorem (the marked Thurstons theorem). We also mention some applications and related results, as well as the notion of deformation spaces of rational maps introduced by A. Epstein.1 The spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 The gluing and the operad structures . . . . . . . . . . . . . . . . . . . . . . 11 3 Framed little discs and the Gerstenhaber and BV Structures . . . . . . . . . 23 4 Moduli space, the Sullivan–PROP and (framed) little discs . . . . . . . . . . 35 5 Stops, Stabilization and the Arc spectrum . . . . . . . . . . . . . . . . . . . 40 6 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7 Open/Closed version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

On the hyperbolic length and quasiconformal mappings

Let ϕ : R → S be a K-quasiconformal mapping of a hyperbolic Riemann surface R to another S. It is important to see how the hyperbolic structure is changed by ϕ. S. Wolpert (1979, The length spectrum as moduli for compact Riemann surfaces. Ann. of Math. 109, 323–351) shows that the length of a closed geodesic is quasi-invariant. Recently, A. Basmajian (2000, Quasiconformal mappings and geodesics in the hyperbolic plane, in The Tradition of Ahlfors and Bers, Contemp. Math. 256, 1–4) gives a variational formula of distances between geodesics in the upper half-plane. In this article, we improve and generalize Basmajians result. We also generalize Wolperts formula for loxodromic transformations.

Dirichlet solutions on bordered Riemann surfaces and quasiconformal mappings

AbstractIn this paper, we consider quasiconformal homeomorphisms ϕn ;S0→Sn (n = 1, 2, ...) of a bordered Riemann surfaceS0 and discuss how the Dirichlet solutions

On analytic and geometric properties of Teichmüller spaces

H_{fo\varphi n^{ - 1} }^{S_n }

On the action of the mapping class group for Riemann surfaces of infinite type

for a continuous functionf on ϖS0 vary when the maximal dilatations of ϕn converge to one. Furthermore, we consider the smoothness of Dirichlet solutions for parameters of the quasiconformal deformation.