Thick-thin decomposition for quadratic differentials

Abstract

Quadratic differentials arise naturally in the study of Teichmuller space and Teichmuller geodesics, in particular. A quadratic differential q on a surface S defines a singular Euclidean metric on S. The explicit nature of singular Euclidean metrics often makes estimates simple and combinatorial arguments possible ([GM91], [Mas93]). On the other hand, by the uniformization theorem, there also exists a canonical hyperbolic metric σ in the conformal class of q and the problem of understanding the relationship between the two metrics often arises (see for example [Min92], [Raf05] and [Raf]). There is a well-known decomposition of a hyperbolic surface S into thick and thin parts ([Thu86], [BP92]). The components of the thin part have a simple topology; they are homeomorphic to annuli. The components of the thick part, on the other hand, have bounded geometry, i.e., the diameter and the injectivity radius of a thick piece are bounded both above and below by constants depending on the topology of S only. The geometry of a thick piece Y is coarsely determined by the topology of a short marking (see [MM00] and Section 3.1 below) in Y . For example, the hyperbolic length of a curve α is comparable (up to multiplicative constants depending on the topology of S) to the geometric intersection number between α and a given short marking ([Min93], see also (3)). We are interested in comparing these two metrics restricted to a thick piece Y of S. In general the diameter and the area of Y may be much smaller in the quadratic differential metric compared with the hyperbolic metric (see the example at the end of this paper). However, we show that, after scaling appropriately, Y equipped with the quadratic differential metric has a geometry comparable with the geometry of Y equipped with the hyperbolic metric. The following is our main theorem: