# Construction of nearly hyperbolic distance on punctured spheres

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## Abstract

We define a distance function on the bordered punctured disk \(0<|z|\le 1/e\) in the complex plane, which is comparable with the hyperbolic distance of the punctured unit disk \(0<|z|<1.\) As an application, we will construct a distance function on an *n*-times punctured sphere which is comparable with the hyperbolic distance. We also propose a comparable quantity which is not necessarily a distance function on the punctured sphere but easier to compute.

## Keywords

Hyperbolic metric Punctured sphere Elliptic modular function## Mathematics Subject Classification

Primary 30C35 Secondary 30C55## 1 Introduction

*hyperbolic*. Since the Poincaré metric \(|d\zeta |/(2\,{{\text {Im}}\,}\zeta )\) is invariant under the pullback by analytic automorphisms of \({\mathbb {H}}\) [in other words, under the action of \({{\text {PSL}}}(2,{\mathbb {R}})\)], it descends to a metric on \(\Omega ,\) called the

*hyperbolic metric*of \(\Omega \) and denoted by \(\lambda _\Omega (w)|dw|.\) More explicitly, they are related by the formula \(\lambda _\Omega (p(\zeta ))|p'(\zeta )|=1/(2\,{{\text {Im}}\,}\zeta ),\) where \(p:{\mathbb {H}}\rightarrow \Omega \) is a holomorphic universal covering projection of \({\mathbb {H}}\) onto \(\Omega .\) The quantity \(\lambda _\Omega (w)\) is sometimes called the

*hyperbolic density*of \(\Omega \) and it is independent of the particular choice of

*p*and \(\zeta \in p^{-1}(w).\) Note that \(\lambda _\Omega \) has constant Gaussian curvature \(-4\) on \(\Omega .\) We denote by \(h_\Omega (w_1,w_2)\) the distance function induced by \(\lambda _\Omega ,\) called the

*hyperbolic distance*on \(\Omega \) and the distance is known to be complete. That is, \(h_\Omega (w_1,w_2)=\inf _\alpha \ell _\Omega (\alpha ),\) where the infimum is taken over all rectifiable curves \(\alpha \) joining \(w_1\) and \(w_2\) in \(\Omega \) and

*principle of hyperbolic metric*, which asserts the monotonicity \(\lambda _\Omega (w)\ge \lambda _{\Omega _0}(w)\) and thus \(h_\Omega (w_1,w_2)\ge h_{\Omega _0}(w_1,w_2)\) for \(w, w_1, w_2\in \Omega \subset \Omega _0\) (

*cf.*[15, III.3.6]). For basic facts about hyperbolic metrics, we refer to recent textbooks [11] by Keen and Lakic or a survey paper [6] by Beardon and Minda as well as classical book [3] by Ahlfors. For instance,

It is, however, difficult to obtain an explicit expression of \(\lambda _\Omega (w)\) or \(h_\Omega (w_1,w_2)\) for a general hyperbolic domain \(\Omega \) because a concrete form of its universal covering projection is not known except for several special domains. (It is not easy even for a simply connected domain because it is hard to find its Riemann mapping function in general.) Therefore, as a second choice, estimates of \(\lambda _\Omega (w)\) are useful. Indeed, Beardon and Pommerenke [7] supplied a general but concrete bound for \(\lambda _\Omega (w).\) However, it is still difficult to estimate the induced hyperbolic distance \(h_\Omega (w_1,w_2)\) due to complexity of the fundamental group of \(\Omega .\)

*f*. As a maximal hyperbolic plane domain, the thrice-punctured sphere (the twice-punctured plane) \({\mathbb {C}}_{0,1}:={\mathbb {C}}{\setminus }\{0,1\}\) is particularly important. Letting \(X={\mathbb {C}}_{0,1},\) the inequality (1.2) leads to Schottky’s theorem when \(\Omega \) is the unit disk (

*cf.*[9, 10]), and it leads to the big Picard theorem when \(\Omega \) is a punctured disk (

*cf.*[3, §1–9]). Though the hyperbolic density of \({\mathbb {C}}_{0,1}\) was essentially computed by Agard [2] (see also [18]) and the holomorphic universal covering projection of \({\mathbb {H}}\) onto \({\mathbb {C}}_{0,1}\) is known as an elliptic modular function (see Sect. 2 below and, e.g., [4, p. 279] or [14, Chap. VI]), we do not have any convenient expression of the hyperbolic distance \(h_{{\mathbb {C}}_{0,1}}(w_1,w_2)\) except for special configurations of the points \(w_1,w_2\) (see, for instance, [17, Lemma 3.10], [18, Lemma 5.1]).

*X*which can be computed (or estimated) more easily than the hyperbolic distance \(h_X(w_1,w_2)\) but still comparable with it by concrete bounds. To this end, we first propose a distance function \(D(z_1,z_2)\) on \(0<|z|\le e^{-1}\) given by the formula

*cf.*(3.3) below].

In order to understand the hyperbolic distance \(h_X(w_1,w_2)\) when one of \(w_1,w_2\) is close to a puncture, we should take a careful look at the hyperbolic geodesic nearby the puncture. In Sect. 2, we investigate it by making use of an elliptic modular function as well as the punctured unit disk model. Section 3 is devoted to the study of the function \(D(z_1,z_2).\) In particular, we show that *D* gives a distance on \(0<|z|\le e^{-1}\) and compare with the hyperbolic distances \(h_{{\mathbb {D}}^*}(z_1,z_2)\) of \({\mathbb {D}}^*\) and \(h_{{\mathbb {C}}_{0,1}}(z_1,z_2)\) of \({\mathbb {C}}_{0,1}.\) As an application of the function \(D(z_1,z_2),\) in Sect. 4, we will construct a distance function \(d_X(w_1,w_2)\) on *n*-times punctured spheres *X* which are comparable with the hyperbolic distance \(h_X(w_1,w_2).\) We will summarise our main results in Theorem 4.2. Unfortunately, \(d_X(w_1,w_2)\) is not very easy to compute because we have to take an infimum in the definition. In Sect. 5, we introduce yet another quantity \(e_X(w_1,w_2),\) which can be computed without taking an infimum though it is no longer a distance function on *X*. We will show our main result that \(d_X(w_1,w_2)\) and \(e_X(w_1,w_2)\) are both comparable with the hyperbolic distance \(h_X(w_1,w_2)\) in a quantitative way in Sect. 5.

We would finally thank Matti Vuorinen for posing, more than ten years ago, the problem of finding a quantity comparable with the hyperbolic distance on \({\mathbb {C}}_{0,1}\) and for helpful suggestions.

## 2 Hyperbolic geodesics near the puncture

In order to estimate the hyperbolic distance \(h_X(w_1,w_2)\) of a punctured sphere \(X={\widehat{\mathbb {C}}}{\setminus }\{a_1,\dots ,a_n\},\) we have to investigate the behaviour of a hyperbolic geodesic joining two points near a puncture. Here and in what follows, a curve \(\alpha \) joining \(w_1\) and \(w_2\) in a hyperbolic domain \(\Omega \) is called a *hyperbolic geodesic* if \(\ell _\Omega (\alpha )\le \ell _\Omega (\beta )\) whenever \(\beta \) is a curve joining \(w_1\) and \(w_2\) which is homotopic to \(\alpha \) in \(\Omega .\) In particular, \(\alpha \) is called *shortest* if \(\ell _\Omega (\alpha ) =h_\Omega (w_1,w_2).\) Note that the shortest hyperbolic geodesic is not unique in general. Our basic model for that is the punctured disk \({\mathbb {D}}^*.\) In this case, we have precise information about the hyperbolic geodesic.

### Lemma 2.1

### Proof

First note that the function \(p(\zeta )=e^{\pi i\zeta }\) is a universal covering projection of the upper half-plane \({\mathbb {H}}\) onto \({\mathbb {D}}^*\) with period 2. We may assume that \(z_1\in (0,1)\) and \(\theta =({{\text {arg}}\,}z_2)/\pi \in (0,1].\) Then \(p(i\eta _1)=z_1,~ p(i\eta _2+\theta )=z_2,\) and \(h_{\mathbb {H}}(i\eta _1,i\eta _2+\theta )=h_{{\mathbb {D}}^*}(z_1,z_2).\) Let \(\tilde{\beta }\) be the hyperbolic geodesic joining \(\zeta _1=i\eta _1\) and \(\zeta _2=i\eta _2+\theta \) in \({\mathbb {H}}.\) Recall that \(\tilde{\beta }\) is part of the circle orthogonal to the real axis. If we fix \(\eta _1,\) the possible largest imaginary part of \(\tilde{\beta }\) is attained when \(\eta _2=\eta _1\) and \(\theta =1.\) Therefore, \(\,{{\text {Im}}\,}\zeta \le \eta _1+\delta \) for \(\zeta \in \tilde{\beta },\) where \(\delta =\sqrt{\eta _1^2+1/4}-\eta _1.\) Hence, we conclude that \(\beta =p(\tilde{\beta })\) is contained in the closed annulus \(e^{-\pi (\eta _1+\delta )}=e^{-\pi \delta }|z_1|\le |z|\le |z_2|\) as required. \(\square \)

By the proof, we observe that the above constant \(\delta \) is sharp. Note here that \(\delta \) is decreasing in \(\eta _1\) and that \(0<\delta <\frac{1}{8\eta _1}.\)

In the above theorem, we see that the subdomain \({\mathbb {D}}^*(0,\rho )\) of \({\mathbb {D}}^*\) with \(0<\rho <1\) is hyperbolically convex. This is also true in general. Indeed, the following result is a special case of Minda’s reflection principle [13] (apply his Theorem 6 to the case when \(\overline{R}=\overline{\Delta }\)).

### Lemma 2.2

Let \(\Omega \) be a hyperbolic subdomain of \({\mathbb {C}}\) and let \(\Delta \) be an open disk centered at a point \(a\in {\mathbb {C}}{\setminus }\Omega .\) Suppose that \(I(\Omega {\setminus }\Delta )\subset \Omega ,\) where *I* denotes the reflection in the circle \(\partial \Delta .\) Then \(\Delta \cap \Omega \) is hyperbolically convex in \(\Omega .\)

In particular, we have

### Corollary 2.3

Let \(\Delta \) be an open disk centered at a puncture *a* of a hyperbolic punctured sphere \(X\subset {\mathbb {C}}\) with \(\Delta ^*=\Delta {\setminus }\{a\}\subset X.\) Then \(\Delta ^*\) is hyperbolically convex in *X*.

*J*and related functions. The covering transformation group is the modular group \(\Gamma (2)\) of level 2; namely, \(\Gamma (2)=\{A\in {{\text {SL}}}(2,{\mathbb {Z}}): A\equiv I~{{\text {mod}}\,}2\}/\{\pm I\}.\) Since

*J*is periodic with period 2, it factors into \(J=Q\circ p,\) where \(p(\zeta )=e^{\pi i\zeta }\) as before and

*Q*is an intermediate covering projection of \({\mathbb {D}}^*\) onto \({\mathbb {C}}_{0,1}.\) Since \(J(\zeta )\rightarrow 0\) as \(\eta =\,{{\text {Im}}\,}\zeta \rightarrow +\infty ,\) the origin is a removable singularity of

*Q*(

*z*) and indeed the function

*Q*(

*z*) has the following representations (see [14, VI.6] or [11, Theorem 14.2.2]):

*Q*(

*z*) has been recently used to improve coefficient estimates of univalent harmonic mappings on the unit disk in Abu Muhanna, Ali and Ponnusamy [1]. By its form,

*Q*(

*z*) is locally univalent at \(z=0.\) In fact, we can show the following.

### Lemma 2.4

The function *Q*(*z*) is univalent on the disk \(|z|<e^{-\pi /2}\approx 0.20788.\) The radius \(e^{-\pi /2}\) is best possible.

### Proof

Suppose that \(Q(z_1)=Q(z_2)\) for a pair of points \(z_1,z_2\) in the disk \(|z|<e^{-\pi /2}.\) Take the unique point \(\zeta _l=\xi _l+i\eta _l\in p^{-1}(z_l)\subset {\mathbb {H}}\) such that \(-1<\xi _l\le 1\) and \(\eta _l>1/2\) for \(l=1,2.\) We now recall the well-known fact that \(\omega =\{\zeta \in {\mathbb {H}}: -\,1<{{\text {Re}}\,}\zeta \le 1, |\zeta +1/2|<1/2, |\zeta -1/2|\le 1/2\}\) is a fundamental set for the modular group \(\Gamma (2)\) of level 2 (see [4, Chap.7, §3.4]). In other words, \(J(\omega )={\mathbb {C}}_{0,1}\) and no pairs of distinct points in \(\omega \) have the same image under the mapping *J*. We now note that \(\zeta _l\in \omega \) for \(l=1.2.\) Since \(J(\zeta _1)=Q(z_1)=Q(z_2)=J(\zeta _2),\) we conclude that \(\zeta _1=\zeta _2.\) Hence, \(z_1=z_2,\) which implies that *Q*(*z*) is univalent on \(|z|<e^{-\pi /2}.\) To see sharpness, we consider the pair of points \(\zeta _1=(1+i)/2\) and \(\zeta _2=(-1+i)/2.\) Since \(\zeta _2=T(\zeta _1)\) for the modular transformation \(T(\zeta )=\zeta /(-2\zeta +1)\) in \(\Gamma (2),\) we have \(J(\zeta _1)=J(\zeta _2).\) Thus \(Q(ie^{-\pi /2})=Q(-ie^{-\pi /2}).\) \(\square \)

We remark that the formula \(Q(ie^{-\pi /2})=2\) is valid. Indeed, by recalling the functional identity \(Q(-z)=Q(z)/(Q(z)-1)\) (see [14, (92) in p. 328]), we have \(Q(ie^{-\pi /2})=Q(-ie^{-\pi /2}) =Q(ie^{-\pi /2})/(Q(ie^{-\pi /2})-1)\), which implies \(Q(ie^{-\pi /2})=2.\)

### Lemma 2.5

### Proof

*r*and \(\mu \) can be computed by the formula

*Q*. We may assume that \(|z_1|\le |z_2|.\) It follows from Lemma 2.1 that \(\beta \) is contained in the annulus \(|z_1|e^{-\pi \delta }\le |z|\le |z_2|,\) where \(\delta \) is given in the lemma with \(\eta _1=-(\log r)/\pi \le -(\log |z_1|)/\pi .\) Hence, by (2.1), the curve \(\alpha =Q(\beta )\) is contained in the annulus

As an immediate consequence, we obtain the following result.

### Corollary 2.6

Let \(0<\sigma \le \rho \le 4e^{-\pi /2}.\) A shortest hyperbolic geodesic \(\alpha \) joining \(z_1,z_2\) in \({\mathbb {C}}_{0,1}\) with \(|z_1|,|z_2|\ge \sigma \) does not intersect the disk \(\Delta ={\mathbb {D}}(0,\sigma e^{-K}),\) where \(K=K(\rho )>0\) is the constant in Lemma 2.5.

### Proof

Suppose that \(\alpha \) intersects the disk \(\Delta .\) Then we can choose a subarc \(\alpha _0\) of \(\alpha \) such that \(\alpha _0\) intersects \(\Delta \) and that both endpoints \(w_1,w_2\) of \(\alpha _0\) have modulus \(\sigma .\) Applying Lemma 2.5 to \(\alpha _0\) yields a contradiction.

When \(\rho =\rho _0=e^{-1},\) we compute \(r_0=e^{1-\pi }\big (8-e^{\pi /2-1}-4\sqrt{4-e^{\pi /2-1}}\big ) \approx 0.0301441\) and \(\mu _0=r_0 e^{\pi /2}\approx 0.145007.\) Also, we have \(\delta _0=\sqrt{\eta _0^2+1/4}-\eta _0\approx 0.107007\) with \(\eta _0=-(\log r_0)/\pi \approx 1.11465.\) Then \(K_0=K(e^{-1})=\pi \delta _0+2\log [(1+\mu _0e^{-\pi \delta _0})/(1-\mu _0)]\approx 0.846666\) and \(e^{K_0}\approx 2.33186.\) Thus we obtain the following statement as a special case of Corollary 2.6.

### Corollary 2.7

Let \(z_1, z_2\) be two points in \({\mathbb {C}}_{0,1}\) with \(|z_1|, |z_2|\ge \sigma \) for some number \(\sigma \in (0, e^{-1}].\) Then, a shortest hyperbolic geodesic \(\alpha \) joining \(z_1,z_2\) in \({\mathbb {C}}_{0,1}\) does not intersect the disk \(|z|<\sigma e^{-K_0},\) where \(K_0=K(e^{-1})\approx 0.846666.\)

## 3 Basic properties of the distance function \(D(w_1,w_2)\)

First we show the following result in the present section. Let \(E^*=\{z: 0<|z|\le e^{-1}\}.\)

### Lemma 3.1

The function \(D(z_1, z_2)\) given by (1.3) is a distance function on the set \(E^*.\)

### Proof

Next we compare \(D(z_1,z_2)\) with the hyperbolic distance \(h_{{\mathbb {D}}^*}(z_1,z_2)\) of \({\mathbb {D}}^*\) on the set \(E^*.\)

### Theorem 3.2

We remark that \(4/\pi \approx 1.27324.\)

### Proof

*z*|. Noting the inequality \(|dz|\ge rd\theta \) for \(z=re^{i\theta },\) we have

## 4 Construction of a distance function on *n*-times punctured sphere

*n*-times punctured sphere \(X={\widehat{\mathbb {C}}}{\setminus }\{a_1,\dots , a_n\}.\) As we noted,

*X*is hyperbolic if and only if \(n\ge 3.\) Thus we will assume that \(n\ge 3\) in the sequel. After a suitable Möbius transformation, without much loss of generality, we may assume that \(0,1,\infty \in {\widehat{\mathbb {C}}}{\setminus } X\) and \(a_n =\infty \). Let

*j*,

*k*. Note also that \(\Delta _j\subset X\) for \(j=1,\dots , n.\) Finally, let \(W=X{\setminus } (E^*_1\cup \dots \cup E^*_n)\).

*X*. Set

### Lemma 4.1

\(d_X\) is a distance function on *X*.

### Proof

*j*,

*k*,

*l*. Then,

*D*is defined by (1.3). Then we can see that \(d_{{\mathbb {C}}^*}\) is a distance function on \({\mathbb {C}}^*.\) The asymptotic behaviour of \(d_{{\mathbb {C}}^*}\) near the punctures are rather different from that of the quasi-hyperbolic distance

*q*on \({\mathbb {C}}^*\) since \(q(w_1,w_2)=|\log |w_1|-\log |w_2||+O(1)\) as \(w_1,w_2\rightarrow 0\) (see [12] for instance).

Our main result in the present paper is the following. In the next section, we will prove it in a stronger form (Theorem 5.1).

### Theorem 4.2

The distance function \(d_X(w_1, w_2)\) given in (4.2) on the *n*-times punctured sphere \(X={\widehat{\mathbb {C}}}{\setminus }\{a_1,\dots ,a_n\} \subset {\mathbb {C}}_{0,1}\) is comparable with the hyperbolic distance \(h_X(w_1,w_2)\) on *X*.

## 5 Proof of Theorem 4.2

*X*. On the other hand, it is not easy to compute the exact value of \(d_X(w_1,w_2)\) for a given pair of points \(w_1, w_2\in X.\) The following quantity can be a good substitute of \(d_X(w_1,w_2)\) because it is computed easily, though it is not necessarily a distance function:

### Theorem 5.1

### Proof

*j*. We can deal with the other cases similarly and thus we will omit it. We further assume, for a moment, that \(j\ne n.\) By definition,

*L*by the shorter component of \(\partial E_k{\setminus } L\) for each such

*k*. The resulting curve will be denoted by \(L'.\) It is obvious from construction that the Euclidean length of \(L'\) is bounded by \(\pi |\zeta _0-w_2|/2.\) Therefore, by (5.1),

*g*maps \(\Delta _j\) conformally onto \({\mathbb {D}}^*,\) the principle of the hyperbolic metric leads to the following:

We next show the inequality \(e_X(w_1,w_2)\le N_2h_X(w_1,w_2).\) We consider several cases according to the location of \(w_1,w_2.\)

*Case (i)*\(w_1, w_2\in E^*_j:\) We first assume that \(j\ne n\) and choose \(k\ne j\) so that \(\tilde{\rho }_j=|a_k-a_j|.\) Let \(X_1={\mathbb {C}}{\setminus }\{a_j,a_k\}.\) Then \(X_1\supset X\) and \(g(w)=(w-a_j)/(a_k-a_j)\) maps \(X_1\) conformally onto \({\mathbb {C}}_{0,1}.\) Set \(z_1=g(w_1)\) and \(z_2=g(w_2).\) By Theorem 3.2, we obtain

*Case (ii)*\(w_1\in E^*_j\) and \(w_2\in W:\) Let \(\zeta \) be the intersection point of the line segment \([w_1,w_2]\) with the boundary circle \(\partial E_j.\) It is thus enough to show the inequalities

*X*conformally into \({\mathbb {C}}_{0,1}.\) (When \(j=1,\) we set \(g(w)=1/w.\)) Put \(z_l=g(w_l)\) for \(l=1,2\) and set \(X_1 =g^{-1}({\mathbb {C}}_{0,1})\) which contains

*X*. We consider a shortest hyperbolic geodesic \(\alpha \) joining \(w_1\) and \(w_2\) in \(X_1.\) Note that \(\ell _{X_1}(\alpha )=h_{X_1}(w_1,w_2)\le h_X(w_1,w_2)\) by the principle of hyperbolic metric. In order to complete the proof, we need to analyse the location of the geodesic \(\alpha .\) Since \(|w_l|\le \rho _n,~l=1,2,\) the points \(z_l\) satisfy \(|z_l|=|a_j/w_l|\ge \sigma ,\) where \(\sigma =|a_j|/\rho _n\le \tilde{\rho }_n/\rho _n\le e^{-1}.\) (When \(j=1,\) \(\sigma :=1/\rho _n\le e^{-1}\) because \(1\notin X.\)) We now apply Corollary 2.7 to see that \(g(\alpha )\) is contained in the set \(|z|\ge \sigma e^{-K_0},\) where we recall that \(K_0\approx 0.85.\) Thus, \(\alpha \) lies in the set \(|w|\le |a_j|e^{K_0}/\sigma =\rho _ne^{K_0}.\) We note also that \(\sigma =|a_j|/\rho _n\ge \tilde{\rho }_1/\rho _n=e\rho _1/\rho _n.\) Hence, by (3.5), we have the lower estimate

We have now shown the inequality \(e_X(w_1,w_2)\le N_2' h_X(w_1,w_2)\) in this case, where \(N_2'=B_1+B_2.\)

*Case (iii)* \(w_1\in W, w_2\in E_j^*\): This is essentially same as case (ii).

*Case (iv)* \(w_1,w_2\in W\): Similarly, we obtain \(|w_1-w_2|\le B_2h_X(w_1,w_2)\) with the same constant \(B_2\) as in case (ii).

*Case (v)*\(w_1\in E_j^*\) and \(w_2\in E_k^*\) with \(j\ne k\): Then, by definition,

*X*. Let \(\zeta _l'\) be an intersection point of \(\alpha \) with \(\beta _l\) for \(l=1,2.\)

Summarising all the cases, we now conclude that the right-most inequality in the assertion of the theorem holds with the choice \(N_2=\max \{N_2',N_2''\}.\) \(\square \)

## Notes

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