Uniformisation of the twice–punctured disc - problems of confluence

Hempel, Joachim A.; Smith, Simon J.

doi:10.1017/s0004972700003294

Cited by 15 publications

(13 citation statements)

References 4 publications

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“…7]. For example, it would be interesting to perform a detailed study of cosmological trajectories for some triply-connected non-elementary planar surfaces such as the twice-punctured disk [36][37][38][39] and once-punctured annulus [40].…”

Section: Conclusion and Further Directionsmentioning

confidence: 99%

Generalizedα-Attractor Models from Elementary Hyperbolic Surfaces

Babalic

Lazaroiu

2018

Advances in Mathematical Physics

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We consider generalized -attractor models whose scalar potentials are globally well-behaved and whose scalar manifolds are elementary hyperbolic surfaces. Beyond the Poincaré disk D, such surfaces include the hyperbolic punctured disk D * and the hyperbolic annuli A( ) of modulus = 2 log > 0. For each elementary surface, we discuss its decomposition into canonical end regions and give an explicit construction of the embedding into the Kerekjarto-Stoilow compactification (which in all three cases is the unit sphere), showing how this embedding allows for a universal treatment of globally well-behaved scalar potentials upon expanding their extension in real spherical harmonics. For certain simple but natural choices of extended potentials, we compute scalar field trajectories by projecting numerical solutions of the lifted equations of motion from the Poincaré half plane through the uniformization map, thus illustrating the rich cosmological dynamics of such models.

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Section: Conclusion and Further Directionsmentioning

confidence: 99%

Generalizedα-Attractor Models from Elementary Hyperbolic Surfaces

Babalic

Lazaroiu

2018

Advances in Mathematical Physics

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“…The point ζ can be determined from the geometry of D, specifically the length of the closed hyperbolic geodesic separating the points p 1 and p 2 from the boundary of D 0 . This example is taken from [9] where a more detailed discussion is available.…”

Section: Preliminariesmentioning

confidence: 99%

Compact composition operators with symbol a universal covering map

Jones

2015

Journal of Functional Analysis

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In this paper we study composition operators, C φ , acting on the Hardy spaces that have symbol, φ, a universal covering map of the disk onto a finitely connected domain of the form D 0 \{p 1 , · · · , p n }, where D 0 is simply connected and p i , i = 1, . . . , n, are distinct points in the interior of D 0 . We consider, in particular, conditions that determine compactness of such operators and demonstrate a link with the Poincare series of the uniformizing Fuchsian group. We show that C φ is compact if, and only if φ does not have a finite angular derivative at any point of the unit circle, thereby extending the result for univalent and finitely multivalent φ.

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“…As an application of the calculation of the accessory parameters of mappings of s.c.q.s, we consider the triply-connected plane domain G a = D − {−a, a} where the value 0 < a < 1 is fixed. The universal covering map H: D → G a has been studied in [9,10,22]. We can assume H(0) = 0 and H ′ (0) > 0.…”

Section: Universal Cover Of a Doubly-punctured Diskmentioning

confidence: 99%

Conformal Mapping of Circular Quadrilaterals and Weierstrass Elliptic Functions

Brown

Porter

2012

Comput. Methods Funct. Theory

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Numerical and theoretical aspects of conformal mappings from a disk to a circular-arc quadrilateral, symmetric with respect to the coordinate axes, are developed. The problem of relating the accessory parameters (prevertices together with coefficients in the Schwarzian derivative) to the geometric parameters is solved numerically, including the determination of the parameters for univalence. The study involves the related mapping from an appropriate Euclidean rectangle to the circulararc quadrilateral. Its Schwarzian derivative involves the Weierstrass ℘-function, and consideration of this related mapping problem leads to some new formulas concerning the zeroes and the images of the half-periods of ℘.

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Uniformisation of the twice–punctured disc - problems of confluence

Cited by 15 publications

References 4 publications

Generalizedα-Attractor Models from Elementary Hyperbolic Surfaces

Generalizedα-Attractor Models from Elementary Hyperbolic Surfaces

Compact composition operators with symbol a universal covering map

Conformal Mapping of Circular Quadrilaterals and Weierstrass Elliptic Functions

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