Toy Teichmüller spaces of real dimension 2: the pentagon and the punctured triangle

Abstract

We study two 2-dimensional Teichmüller spaces of surfaces with boundary and marked points, namely, the pentagon and the punctured triangle. We show that their geometry is quite different from Teichmüller spaces of closed surfaces. Indeed, both spaces are exhausted by regular convex geodesic polygons with a fixed number of sides, and their geodesics diverge at most linearly.

Zeros of the isomonodromic tau functions in constructive conformal mapping of polycircular arc domains: the n-vertex case

“…Table 1 shows the numerical values for the prevertex locations and the accessory parameters associated with the regular pentalateral for r = 0.7. The values shown are calculated using three different schemes: a numerical construction of simply connected polycircular-arc domains devised by Howell [31] based on direct numerical quadrature, the new τ function method presented here and, for the prevertices only, the theoretical values (144) found in [30]. The values for w 1 and w 2 agree to machine precision with the exact values given in (144).…”

“…A parameter r governs the curvature of the circular-arc edges. In this case, it turns out that the values for the prevertices can be obtained in exact form by consideration of the action of the symmetry group of the pentagon D 5 [30]. The values turn out to be independent of r:…”

“…The values for w 1 and w 2 agree to machine precision with the exact values given in (144). Note that the action of D 5 considered in [30] does not pin down the values of the accessory parameters β 1 and β 2 and we are unaware of an alternative method of computing them.…”

A qualitative description of the horoboundary of the Teichmüller metric

“…Proof. Assuming that there is some foliation F with a recurrent leaf to some part of S we get a contradiction, as explained in the proof of [8,Lemma 4.1]. Hence, each indecomposable foliation is a proper arc.…”

“…We say that a subset of a boundary component is a boundary arc if it is homeomorphic to an open interval or a circle, does not contain marked points and, if it is homeomorphic to an open interval, it is delimited by marked points. Repeating the argument by Chen, Chernov, Flores, Fortier Bourque, Lee and Yang from [8] to a more general setting, we get the following lemma, which we shall use to solve the simpler cases.…”