Teichmuller Space Of Riemann Surfaces Research Articles

On the length spectrum metric in infinite dimensional Teichmüller spaces

We consider the length spectrum metric dL in infinite dimensional Teichmuller space T(R0). It is known that dL defines the same topology as that of the Teichmuller metric dT on T(R0) if R0 is a topologically finite Riemann surface. In 2003, Shiga proved that dL and dT define the same topology on T(R0) if R0 is a topologically infinite Riemann surface which can be decomposed into pairs of pants such that the lengths of all their boundary components except punctures are uniformly bounded by some positive constants from above and below. In this paper, we extend Shiga's result to Teichmuller spaces of Riemann surfaces satisfying a certain geometric condition.

Extension of Geodesic Algebras to Continuous Genus

Using the Penner--Fock parameterization for Teichmuller spaces of Riemann surfaces with holes, we construct the string-like free-field representation of the Poisson and quantum algebras of geodesic functions in the continuous-genus limit. The mapping class group acts naturally in the obtained representation.

Asymptotic Flatness of the Weil-Petersson Metric on Teichmüller Space

In this paper, we show that there is no negative upper bound for the sectional curvature of Teichmuller space of Riemann surfaces with the Weil–Petersson metric. Mathematics Subject Classifications (2000). 32G15, 53C21, 30F60.

Harmonic diffeomorphisms and Teichmüller theory

Teichmuller spaces of Riemann surfaces usually are treated by using quasi conformal mappings. We prove the existence of a harmonic diffeomorphism between punctured surfaces. We take this to build up a Teichmuller theory and give a new proof of Teichmuller's theorem for Riemann surfaces of finite type.

On the holomorphy of maps from a complex to a real manifold

Let/: X -» Ybe a C 1-submersion from a complex manifold Xto a real C1-manifold Y. One main object of this paper is to prove two sets of necessary and sufficient conditions which will guarantee that Y can be equipped with a complex structure making / holomorphic. We provide a generic counterexample to show the essential nature of the conditions we establish. The second set of conditions (Theorem 3) apply even when X is a complex Banach manifold. This theorem is then used to prove the existence of the natural complex structure on the Teichmuller spaces of Riemann surfaces of finite type.

A rough fundamental domain for Teichmüller spaces

Let T(S) be the Teichmuller space of Riemann surfaces of finite type and let M(S) be the corresponding modular group. In [11] we described T(S) in terms of real analytic parameters. In this paper we determine a subspace R(S) of T(S) which is a fundamental for M (S) acting on T(S). The construction of R(S) is a generalization of the constructions in [14] and [15]. The previous constructions depended heavily upon an analysis of the action of the elements of M (S) on parameters of T(S) corresponding to disjoint closed geodesies on S, and on a theorem of Bers [2] which gives bounds for the lengths of these curves. In the general case, the disjoint closed geodesies of Bers' theorem no longer always correspond directly to the parameters. Hence we must carefully study how their lengths are related to the parameters. In §1 we outline the basic preliminary notions relating hyperbolic geometry, Fuchsian groups and Teichmuller spaces. In §§2 and 3 we give the constructions of Teichmuller space and of a fundamental domain for the action of the modular group in the simplest cases; that is, where S has type (0; 3) and (1; 1). In §4 we give a detailed discussion of the Teichmuller space and of the fundamental domain in the case (0; 4). These constructions are the heart of the general constructions which follow. In §5 we discuss the special case of surfaces of genus 2 and state Bers* theorem. In §6 we give the construction of Teichmuller space in general. In §7 we analyze the topologically distinct sets of mutually disjoint geodesies which occur in Bers' theorem and determine their relationship to the moduli curves. Finally, in §8 we put all the pieces of the construction together and determine the rough fundamental domain. In §9 we use this construction to affirmatively settle a conjecture of Bers [3].