THICK-THIN DECOMPOSITION FOR QUADRATIC DIFFERENTIALS

By Kasra Rafi. 2014

61 citations recorded by the source corpus.

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:

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7f8e5909b871d654820a15fe0ba8b75417c9ace1

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Lattice Point Asymptotics and Volume Growth on Teichmuller space. (2006, background): “Lemma 8.1 (Rafi, [ 22 ]) Let q be a holomorphic quadratic differential on a surface S, and let σ be the hyperbolic metric on S in the same conformal class as q. Choose C ≫ 1, and let the ǫi, 1 ≤ i ≤ m and the complexes Ki be as in the previous paragraph.”
Comparison between Teichmüller and Lipschitz metrics (2005, background): “Since the moduli space of the thick part is compact, we know that the hyperbolic lengths of curves in σ and τ are proportional to their quadratic differential lengths in qa and qb, respectively (see [10] for a more general discussion).”
Hyperbolicity in Teichmüller space (2010, background): “Hyperbolicity in Teichmüller space 3037 plan, nearly identical to that of [17], is to fill all components of @Y with locally flat once-punctured disks.”