Hyperbolic Lengths of Geodesics Surrounding Two Punctures

Abstract

ABSTRACT. For the plane regions Oi = {\z\ < R,z / 0,1} with R > 1, and 0.2 = C \ {0, l,p} with \p\ = R > 1, we describe, as R -» oo, the hyperbolic lengths of the geodesies surrounding 0 and 1. Upper and lower bounds for the lengths are also stated, and these results are used to obtain inequalities, which are precise in a certain sense, for the length of the geodesic surrounding 0 and 1 in an arbitrary plane region 0 satisfying 121 c 0 C 0,2.

Uniformisation of the twice–punctured disc - problems of confluence

“…Thus t/i(z) and n = -oo y 2 (z) must be linearly dependent, and so there exists a j= 0 such that A_ n +aA n = 0 for all n . From this it follows that \a\ = 1, and so the function z 1 -*-1 / 2 V^ a n z 2n , where n = -oo a n = ia~1/ 2 A n for all n, is a solution to (2.12) in p < \z\ < p~1 with the property that the coefficients a n satisfy (2.6(ii)). The relation (2.6(i)) is then immediate, while (2.6(iii)) follows from (2.13).…”

“…Hempel and S.J. Smith [2] There are essentially three parameters whose determination is of particular concern. Firstly, if T denotes the homotopy class in fi p of a circle separating {±p} from {z : \z\ -1} , then J-contains a unique hyperbolic geodesic.…”

“…Firstly, if T denotes the homotopy class in fi p of a circle separating {±p} from {z : \z\ -1} , then J-contains a unique hyperbolic geodesic. The length L of this geodesic is the parameter of interest, not only because of its defining geometric interpretation, but also because it is related to trace T, the trace of the hyperbolic covering transformation T of <f> determined by T, according to |trace T\ = 2 cosh ( -As a preliminary result, we remark that Theorem 1 of [2] is equivalent to Although the inverse r = T(Z) of a universal cover of fi p by U is multiple-valued, the Schwarzian derivative {r, z} of T , defined by is a single-valued meromorphic function given by…”

Generalizedα-Attractor Models from Elementary Hyperbolic Surfaces

“…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].…”

Uniformization of a Once-Punctured Annulus

“…It is, however, difficult to find an explicit form of the holomorphic universal cover π or the covering group G, except for several special cases (see e.g., [8,17]). For a twice-punctured unit disk, Hempel and Smith [9][10][11] considered the uniformization problem and the hyperbolic metric, and Beardon [4] provided five parameters to characterize the twice-punctured disk via its hyperbolic structure and complex structure. Nevanlinna [14, I.3, I.4] introduced a method to regard the puncture as the extremal case when a boundary curve shrinks to a single point.…”