Carathéodory metric on some generalized Teichmüller spaces (original) (raw)

On local comparison between various metrics on Teichmüller spaces

Geometriae Dedicata, 2012

There are several Teichmüller spaces associated to a surface of infinite topological type, after the choice of a particular basepoint ( a complex or a hyperbolic structure on the surface). These spaces include the quasiconformal Teichmüller space, the length spectrum Teichmüller space, the Fenchel-Nielsen Teichmüller space, and there are others. In general, these spaces are set-theoretically different. An important question is therefore to understand relations between these spaces. Each of these spaces is equipped with its own metric, and under some hypotheses, there are inclusions between these spaces. In this paper, we obtain local metric comparison results on these inclusions, namely, we show that the inclusions are locally bi-Lipschitz under certain hypotheses. To obtain these results, we use some hyperbolic geometry estimates that give new results also for surfaces of finite type. We recall that in the case of a surface of finite type, all these Teichmüller spaces coincide setwise. In the case of a surface of finite type with no boundary components (and possibly with punctures), we show that the restriction of the identity map to any thick part of Teichmüller space is globally bi-Lipschitz with respect to the length spectrum metric and the classical Teichmüller metric on the domain and on the range respectively. In the case of a surface of finite type with punctures and boundary components, there is a metric on the Teichmüller space which we call the arc metric, whose definition is analogous to the length spectrum metric, but which uses lengths of geodesic arcs instead of lengths of closed geodesics. We show that the restriction of the identity map restricted to any "relative thick" part of Teichmüller space is globally bi-Lipschitz, with respect to any of the three metrics: the length spectrum metric, the Teichmüller metric and the arc metric on the domain and on the range.

Some metric properties of the Teichmüller space of a closed set in the Riemann sphere

Annales Academiae Scientiarum Fennicae Mathematica

Let E be an infinite closed set in the Riemann sphere, and let T (E) denote its Teichmüller space. In this paper, we study some metric properties of T (E). We prove Earle's form of Teichmüller contraction for T (E), holomorphic isometries from the open unit disk into T (E), extend Earle's form of Schwarz's lemma for classical Teichmüller spaces to T (E), and finally study complex geodesics and unique extremality for T (E).

On Teichmüller’s metric and Thurston’s asymmetric metric on Teichmüller space

Handbook of Teichmüller Theory, Volume I, 2007

In this chapter, we review some elements of Teichmüller's metric and of Thurston's asymmetric metric on Teichmüller space. One of our objectives is to draw a parallel between these two metrics and to stress on some differences between them. The results that we present on Teichmüller's metric are classical, whereas some of the results on Thurston's asymmetric metric are new. We also discuss some open problems.

On the analytic structure of certain infinite dimensional Teicmüller spaces

Nagoya Mathematical Journal, 1996

It is well known since long time that quasiconformally different finite Riemann surfaces give rise to biholomorphically nonequivalent Teichmüller spaces except for a few obvious cases (cf. [R], [E-K]). This is deduced as an application of Royden’s theorem asserting that the Teichmüller metric is equal to the Kobayashi metric. For the case of infinite Riemann surfaces, however, it is still unknown whether or not the corresponding result holds, although it has been shown by F. Gardiner [G] that Royden’s theorem is also valid for the infinite dimensional Teichmüller spaces. On the other hand, recent activity of several mathematicians shows that the infinite dimensional Teichmüller spaces are interesting objects of complex analytic geometry (cf. [Kru], [T], [N], [E-K-K]). Therefore, based on the generalized form of Royden’s theorem, one might well look for further insight into Teichmüller spaces by studying the above mentioned nonequivalence question.

Convex-like properties of the Teichmüller metric

Contemporary Mathematics, 1999

where · ∞ denotes the L ∞ norm; note that |ν(z)|/(1 − |µ(z)| 2 ) is the Poincaré length of the tangent vector ν(z) at the point µ(z) ∈ ∆. The Teichmüller distance on M is just the † Both authors have been partially supported by a grant of GNSAGA, Consiglio Nazionale delle Ricerche, Italy.

Classifying Complex Geodesics for the Carathéodory Metric on Low-Dimensional Teichmüller Spaces

Journal d'Analyse Mathématique, 2020

It was recently shown that the Carathéodory and Teichmüller metrics on the Teichmüller space of a closed surface do not coincide. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichmüller disk generated by a differential with no odd-order zeros. Our aim is to classify Teichmüller disks on which the two metrics agree, and we conjecture that the Carathéodory and Teichmüller metrics agree on a Teichmüller disk if and only if the Teichmüller disk is generated by a differential with no odd-order zeros. Using dynamical results of Minsky, Smillie, and Weiss, we show that it suffices to consider disks generated by Jenkins-Strebel differentials. We then prove a complex-analytic criterion characterizing Jenkins-Strebel differentials which generate disks on which the metrics coincide. Finally, we use this criterion to prove the conjecture for the Teichmüller spaces of the five-times punctured sphere and the twice-punctured torus. We also extend the result that the Carathéodory and Teichmüller metrics are different to the case of compact surfaces with punctures.

Some metrics on Teichmüller spaces of surfaces of infinite type

Transactions of the American Mathematical Society, 2011

Unlike the case of surfaces of topologically finite type, there are several different Teichmüller spaces that are associated to a surface of topological infinite type. These Teichmüller spaces first depend (set-theoretically) on whether we work in the hyperbolic category or in the conformal category. They also depend, given the choice of a point of view (hyperbolic or conformal), on the choice of a distance function on Teichmüller space. Examples of distance functions that appear naturally in the hyperbolic setting are the length spectrum distance and the bi-Lipschitz distance, and there are other useful distance functions. The Teichmüller spaces also depend on the choice of a basepoint. The aim of this paper is to present some examples, results and questions on the Teichmüller theory of surfaces of infinite topological type that do not appear in the setting the Teichmüller theory of surfaces of finite type. In particular, we point out relations and differences between the various Teichmüller spaces associated to a given surface of topological infinite type.

REMARKS ON THE HYPERBOLIC GEOMETRY OF PRODUCT TEICHMULLER SPACES

Let T be a cross product of n Teichmüller spaces of Fuchsian groups, n > 1. From the properties of Kobayashi metric and from the Royden-Gardiner theorem, T is a complete hyperbolic manifold. Each two distinct points of T can be joined by a hyperbolic geodesic segment, which is not in general unique. But when T is finite dimensional or infinite dimensional of a certain kind, then among all such segments there is only one which enjoys a distinguished property: it is obtained from a uniquely determined holomorphic isometry of the unit disc into T .

Approximation of infinite-dimensional Teichmüller spaces

Transactions of the American Mathematical Society, 1984

By means of an exhaustion process it is shown that Teichmuller's metric and Kobavashi's metric arc equal for infinite dimensional Tcichmuller spaces. By the same approximation method important estimates coming from the Reich·Strebel inequality are extended to the infinite dimensional cases. These estimates are used to ,how that Teichmuller's metric is the integral of its infinitesimal form. They are also used to give a sufficient condition for a sequence to be an absolute maximal sequence for the Hamilton functional. Finally, they are used to give a new sufficient condition for a sequence of Beltrami coefficients to converge in the Teichmuller metric.