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

Chen, Yudong; Chernov, Roman; Flores, Marco A.; Bourque, Maxime Fortier; Lee, Seewoo; Yang, Bowen

doi:10.48550/arxiv.1704.04331

Cited by 1 publication

(3 citation statements)

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“…Table I 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).…”

Section: Methodsmentioning

confidence: 71%

“…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.…”

Section: Methodsmentioning

confidence: 99%

“…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:…”

Section: Explicit Examples For 5 Verticesmentioning

confidence: 96%

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

Cunha¹,

Silva²,

Nelson³

et al. 2021

J. Phys. A: Math. Theor.

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The prevertices of the conformal map between a generic, n-vertex, simply connected, polycircular arc domain and the upper half plane are determined by ﬁnding the zeros of an isomonodromic tau function. The accessory parameters of the associated Fuchsian equation are then found in terms of logarithmic derivatives of this tau function. Using these theoretical results a constructive approach to the determination of the conformal map is given and the particular case of 5 vertices is considered in detail. A computer implementation of a construction of the isomonodromic tau function described by Gavrylenko & Lisovyy [Comm. Math. Phys., 363, 2018)] is used to calculate some illustrative examples. A procedural guide to constructing the conformal map to a given polycircular arc domain using the method presented here is also set out.

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Section: Methodsmentioning

confidence: 71%

Section: Methodsmentioning

confidence: 99%