Fiber spaces over Teichmüller spaces for Fuchsian groups with elliptic elements

We determine the biholomorphic fiber preserving isomorphisms of fiber spaces over Teichmüller spaces for Fuchsian groups with elliptic elements. We show that except in some special cases a biholomorphic fiber preserving isomorphism between two Bers fiber spaces is always an allowable mapping. We find that the situation is different for Teichmüller curves, showing that in general there are some other biholomorphic fiber preserving isomorphisms between Teichmüller curves besides the allowable mappings. Research supported by the National Natural Science Foundation of China.     

Self-dual connections,hyperbolic metrics and harmonic mappings on Riemann surfaces

The Sampson-Wolf model of Teichmüller space (using harmonic mappings) is shown to be exactly the same as the more recent Hitchin model (utilizing self-dual connections). Indeed, it is noted how the self-duality equations become the harmonicity equations. An interpretation of the modular group action in this model is mentioned.     

Once-holed tori embedded in Riemann surfaces

Once-holed tori are the most primitive noncompact Riemann surfaces of positive genus, and consitute a partially ordered set, the order being defined in terms of conforaml embeddings. We consider some families of once-holed tori that are conformally embedded in target Riemann surfaces of conformal mappings of a given noncompact Riemann surface of genus one, and establish an analogue of the one-quarter theorem of Koebe. We also investigate families of once-holed tori conformally embedded in a Riemann surface of positive genus.      

On the geometry of infinite dimensional Teichmüller spaces

We prove that in any infinite dimensional Teichmüller space, there exists a minimal geodesic lying in two distinct geodesic disks.     

Ergodicity of mapping class group actions on representation varieties, I. Closed surfaces

We prove that the mapping class group of a closed surface acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group. Received: March 21, 2001     

Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space

The aim of this paper is to develop the theory of a compactification of Teichmüller space given by F. Gardiner and H. Masur, which we call the Gardiner–Masur compactification of the Teichmüller space. We first develop the general theory of the Gardiner–Masur compactification. Secondly, we will investigate the asymptotic behaviors of Teichmüller geodesic rays under the Gardiner–Masur embedding. In particular, we will observe that the projective class of a rational measured foliation G can not be an accumulation point of every Teichmüller geodesic ray under the Gardiner–Masur embedding, when the support of G consists of at least two simple closed curves. Dedicated to Professor Yoichi Imayoshi on the occasion of his 60th birthday.     

Symplectic structure and reduction on the space of Riemannian metrics

The space of Riemannian metrics ${\mathfrak{Met}}MThe space of Riemannian metrics on an oriented compact manifold M of dimension n = 4k − 2 is endowed with a canonical presymplectic structure and a moment map [cf. Ferreiro Pérez and Mu?oz Masqué, Preprint (arXiv: math.DG/0507075)]. The fiber is characterized as the space of solutions to a differential equation. In dimension 2, the symplectic reduction of is analyzed and the construction presented here is compared with that introduced in Donaldson (Fields Medallists’ Lectures, 1997) and Fujiki (Sugaku Expositions 5(2):173–191, 1992). Finally, conformally flat metrics and, for n = 6, K?hler metrics of constant holomorphic sectional curvature are shown to be contained in .      

Baire spaces, Tychonoff powers and the Vietoris topology

In this paper, we show that if the Tychonoff power of a quasi-regular space is Baire, then its Vietoris hyperspace is also Baire. We also provide two examples to show (i) the converse of this result does not hold in general, and (ii) the Baireness of finite powers of a space is insufficient to guarantee the Baireness of its hyperspace.

     

New Families of Complete Caps, and the Asymptotic Size of the Largest Caps, of Quadrics over Prime Fields

A cap of a quadric is a set of its points whose pairwise joins are all chords. Such a cap is complete if it is not part of a larger one. Few examples of complete caps are known except for quadrics in low dimensions. In this paper, we consider the case when the coordinate field is GF(p), with p an odd prime, and construct, in each projective space GF(n,p) with n p – 1 and n – 2(mod p), a cap on one of its nonsingular quadrics. We use this in two ways. Firstly, we combine its size with the recent Blokhuis–Moorhouse upper bound for quadric caps to show that the size of the largest cap of any nonsingular quadric in PG(N,p) is asymptotic to Np – 1/(p – 1) ! as N tends to infinity. Secondly, by establishing situations when our cap is complete, we produce various infinite families of complete quadric caps over GF(p) for each p. Earlier work determined all complete caps of all nonsingular quadrics over GF(2).