Analysis Seminar, Helsinki, 8.9.2008

Hyperbolic metric in

planar domains

Riku Klén

riku.klen@utu.fi

Abstract. We will consider the hyperbolic

metric in planar domains. The talk is based on

the book Hyperbolic Geometry from a Local

Viewpoint by L. Keen and N. Lakic, 2007,

Cambridge University Press.

1

Uniformization theorem

We denote the unit disk by B2 and assume that

all universal covering maps are holomorphic.

We denote by ρ the hyperbolic density in B2.

[ρ(x) = 1/(1 − |x|2)]Theorem. (Riemann mapping theorem.)

Simply connected domain D ( C ⇒ There

exists conformal homeomorphism ϕ from D

onto B2.

Theorem. (Uniformization theorem.) The

universal covering space D̃ of an arbitrary

Riemann surface D is homeomorphic to the

Riemann sphere, the complex plane or B2.

2

Introduction

In this talk we will consider the hyperbolic

metric. Many related metrics have recently

been studied by various authors.

• Deza-Deza: Dictionary of Distances. [DD]

• Papadopoulos and Troyanov: Weak metrics

on Euclidean domains. [PT]

• Aseev, Sychëv and Tetenov:

Möbius-invariant metrics and generalized

angles in Ptolemaic spaces. [AST]

• Herron, Ma and Minda: Möbius invariant

metrics bilipschitz equivalent to the

hyperbolic metric. [HMM]

• Keen and Lakic: Hyperbolic geometry from

a local viewpoint. [KL]

• Betsakos: Estimation of the hyperbolic

metric by using the punctured plane. [B]

• Metrics in connection with quasiconformal

mappings.

3

Definitions

• Domain D ⊂ C, #∂D ≥ 2, is a hyperbolic

domain.

Uniformization theorem ⇒ there exists a

universal covering map π from B2 to any

hyperbolic domain D.

• Let D be a hyperbolic domain, x ∈ D and

t ∈ B2 be such that π(t) = x. Then

hyperbolic density is defined by

ρD(x) =ρ(t)

|π′(t)| .

• Hyperbolic length of a rectifiable path γ is

defined by

ρ(γ) =

∫

γ

ρD(t)|dt|.

• Hyperbolic distance for x, y ∈ D is defined

by

ρ(x, y) = inf ρD(γ)

where the infimum is taken over all

rectifiable curves joining x and y in D.

4

Infinitesimal isometry

A covering (D̃, π) of D is regular if, for all

p ∈ D, every curve γ(t), γ(0) = p, has a lift to

each p̃ ∈ D̃ with π(p̃) = p.

Theorem 1. If g is a regular holomorphic

covering map from hyperbolic domain H onto

plane domain D, then

ρD(g(t))|g′(t)| = ρH(t)

for all t ∈ H.

Proof. π universal covering map from B2 onto

H =⇒ g ◦ π univ. cov. map from B2 onto D.

Any curve γ ∈ D can be lifted to H and then to

B2. For any pre-images t = g−1(z), s = π−1(t)

ρH(t)|π′(s)| = ρ(s) = ρD(g(t))|(g ◦ π)′(s)|

chainrule =⇒

ρH(t)|π′(s)| = ρD(g(t))|g′(t)||π′(s)|

=⇒ ρH(t) = ρD(g(t))|g′(t)|.

5

Theorem 2. Let π be a universal covering map

from B2 onto a plane domain D. If z, w ∈ D

and t ∈ B2 is any pre-image of z, then

ρD(z, w) = min{ρ(t, s) : s ∈ B2, π(s) = w}.

Proof. By definition of the hyperbolic distance

ρD(z, w) = inf{ρ(u, v) : u, v ∈ B2, π(u) = z, π(v) = w}.

The assertion follows since π is continuous.

Theorem 3. For every hyperbolic plane

domain H, (H, ρH) is a complete metric space.

6

Properties of the hyperbolic

metric

From now on we denote by H a hyperbolic

domain and by π the universal covering map

from B2 onto H.

Next we show that hyperbolic density is

infinitesimal form of hyperbolic distance.

Theorem 4. For z ∈ H and t ∈ C

ρH(z, z + t) = |t|ρH(z) + o(t).

Proof. Let a = π−1(z). By Thm. 2 ∃at ∈ B2

such that π(at) = z + t and

ρ(a, at) = ρH(z, z + t). at → a as t → 0 and

therefore

ρH(z, z + t)

|t| =ρ(a, at)

|t| =ρ(a, at)

|at − a|

∣

∣

∣

∣

at − a

t

∣

∣

∣

∣

.

(5)

Since ρ(x, x + t) = |t|ρ(x) + o(t) for x ∈ B2 we

haveρ(a, at)

|at − a| → ρ(a) as t → 0.

7

We also have∣

∣

∣

∣

at − a

t

∣

∣

∣

∣

=

∣

∣

∣

∣

at − a

π(at) − π(a)

∣

∣

∣

∣

→ 1

|π′(a)| as t → 0.

Therefore by (5)

ρH(z, z + t)

|t| → ρ(a)

|π′(a)| = ρH(π(a)) as t → 0.

Corollary 6. The hyperbolic metric in H is

locally equivalent to the Euclidean metric.

Proof. Euclidean distance satisfies

dH(z, z + t) = |t|dH(z) with dH(z) = 1.

8

Theorem 7. Hyperbolic density ρH(z) is a

positive continuous function.

Proof. Let z0 ∈ H and t0 = π−1(z0).

π holomorphic, locally 1-to-1 =⇒ ∃ local

inverse g of π in a neighborhood N of z0. Let

z ∈ N and t = g(z). Now by definition ρH(z)

and the fact that ρ(x) = 1/(1 − |x|2)

ρH(z) =ρ(t)

|π′(t)| = ρ(g(z))|g′(z)| =|g′(z)|

1 − |g(z)|2 > 0.

Definition 8. A curve γ ⊂ H is a geodesic iff

every lift π−1(γ) is a geodesic in B2.

Proposition 9. Let γ ⊂ H be a curve. If for

all x, y, z ∈ γ, y between x and z,

ρH(x, z) = ρH(x, y) + ρH(y, z),

then γ is a geodesic.

9

Existence of geodesics

Theorem 10. For all x, y ∈ H there exists (at

least one) geodesic.

Proof. Thm 2 =⇒ ∃s, t ∈ B2 such that

π(s) = x, π(t) = y and ρ(s, t) = ρH(x, y).

∃ geodesic γ in B2 joining s and t. By Def. 8

π(γ) is a curve joining x and y. Since π

preserves length of curves we have

ρH(x, y) = ρ(t, s) = ρ(γ) = ρH(π(γ)).

10

Pick’s theorem

Theorem. (Pick’s theorem) Let H1 and H2 be

hyperbolic domains. If f is a holomorphic map

from H1 into H2, then

ρH2(f(t))|f ′(t)| ≤ ρH1

(t) (11)

and

ρH2(f(s), f(t)) ≤ ρH1

(s, t) (12)

for all s, t ∈ H1.

Proof. Let πi be the universal covering map

from B2 to Hi for i = 1, 2. We will lift f to a

map g from B2 into B2. Let p = π−11 (t) and

q = π−12 (f(t)) be any pre-images in B2. Pick

arbitrary a ∈ B2 and choose any curve γ ⊂ B2

joining a and p. Lift the curve f(π1(γ)) to a

curve γ′ ⊂ B2 that starts at q. The other

endpoint of γ′ is by definition a. Define g by

the resulting map. [g is welldefined.]

πi holomorphic and 1-to-1 =⇒ g is

holomorphic.

11

Now for all b ∈ B2

f ◦ π1(b) = π2 ◦ g(b)

and therefore

f ′(π1(b))π′1(b) = π′

2(g(b))g′(b).

By Pick’s theorem for ρ we have

ρ(g(b))|g′(b)| ≤ ρ(b) and therefore

ρ(g(b))|f ′(π1(b))π′1(b)| ≤ ρ(b)|π′

2(g(b))|

=⇒ ρ(g(b))

|π′2(g(b))| |f

′(π1(b))| ≤ρ(b)

|π′1(b)|

=⇒ ρH2(π2(g(b)))|f ′(π1(b))| ≤ ρH1

(π1(b))

=⇒ ρH2(f(π1(b)))|f ′(π1(b))| ≤ ρH1

(π1(b)).

π1 surjective =⇒ (11).

12

Let γ ⊂ H1 be a geodesic joining s and t. By

(11) we have

ρH2(f(s), f(t)) ≤ ρH2

(f(γ))

=

∫

f(γ)

ρH2(x)|dx|

=

∫

γ

ρH2(f(x))|f ′(x)||dx|

≤∫

γ

ρH1(x)|dx|

= ρH1(γ) = ρH1

(s, t)

and (12) follows.

Corollary 13. Let H1 and H2 be hyperbolic

domains. If f is a conformal homeomorphism

from H1 onto H2, then

ρH2(f(t))|f ′(t)| = ρH1

(t)

and

ρH2(f(s), f(t)) = ρH1

(s, t)

for all s, t ∈ H1.

Proof. Use Pick’s theorem for f and f−1.

13

Examples I

(simply connected domain)

If H is simply connected, then by Riemann

mapping theorem there exists a conformal

homeomorphism f from B2 onto H and

therefore

ρH(f(t)) =ρ(t)

|f ′(t)| .

• Half-plane H = {z ∈ C : Im z > 0}.Now fH(z) = i(1 + z)/(1 − z),

gH(w) = f−1H

(w) = (w − i)/(w + i) and

ρH(z) = ρ(gH(z))|g′H(z)| =

1

2Im z.

14

• Koebe domain K = C \ (−∞,−1/4].

Now

fK(z) =1

4

(

1 + z

1 − z

)2

− 1

4=

z

(1 − z)2

and

gK(w) = f−1K (w) = 1 − 2/(1 +

√4w + 1).

Therefore

ρK(z) = ρ(gK(z))|g′K(z)| =1

|√

4z + 1|Re√

4z − 1.

• Infinite strip L = {z ∈ C : 0 < Im z < λ}.Now fL(z) = (λ log z)/π,

gL(w) = f−1L (w) = exp(πw/λ) and

ρL(z) =π

2λ sin(

πλIm z

) .

15

Examples II

(punctured disk)

We consider domain B∗ = B2 \ {0}. Function

f(z) = eiz is universal covering map from H

onto B∗. By Theorem 1

ρB∗(f(z))|f(z)| = ρB∗(eiz)|eiz| = ρH(z)

for z ∈ H and

ρB∗(w)|w| = ρH

(

i

log |w|

)

=⇒ ρB∗(w) =1

2|w| 1log |w|

for w ∈ B∗.

16

Examples III

(annulus)

We consider domain

Aa = {z ∈ C : a < |z| < 1}, a ∈ (0, 1). Function

g(w) = eiw maps L to Ae−λ . By Theorem 1

ρAa(g(w))|g′(w)| = ρL(w) =

π

2λ sin(

πλImw

)

for a = e−λ and w ∈ L. Therefore

ρAa(z) =

π

2|z|λ sin(

πλ

log 1|z|

)

for z ∈ Aa.

17

Estimates of hyperbolic densities

In general hyperbolic domain it is impossible to

find explicit formula for hyperbolic density or

distance. Therefore we need to estimate.

Pick’s theorem for holomorphic f , hyperbolic

domains H1, H2 =⇒

• infinitesimal contraction

ρH2(f(t))|f ′(t)| ≤ ρH1

(t)

• global contraction

ρH2(f(s), f(t)) ≤ ρH1

(s, t)

for all s, t ∈ H1.

Hyperbolic density and metric are monotone

with respect to the domain

H1 ⊂ H2 =⇒ ρH1(z) ≥ ρH2

(z)

for all z ∈ H1.

18

Strong contractions

Let H1 and H2 be hyperbolic domains such

that H2 ⊂ H1 and f : H1 → H2 be holomorphic

(f ∈ Hol(H1, H2)). Let g : H2 → H1, g(z) = z

be the inclusion map. By Pick’s theorem for

g ◦ f and f ◦ g we have

ρHj(f(t))|f ′(t)| ≤ ρHj

(t),

ρHj(f(t), f(s)) ≤ ρHj

(t, s)

for all s, t ∈ Hj . We define the global

Hj-contraction constant to be

glHj(f) = sup

z,w∈Hj , z 6=w

ρHj(f(z), f(w))

ρHj(z, w)

and the infinitesimal Hj-contraction constant

to be

lHj(f) = sup

z∈Hj

ρHj(f(z))|f ′(z)|ρHj

(z).

19

Theorem 14. lHj(f) = glHj

(f) ≤ 1 for

j = 1, 2.

Proof. Let z, w ∈ H1 and γ ⊂ H1 be a geodesic

joining z and w. Now

ρH1(f(z), f(w)) ≤ ρH1

(f(γ))

≤ lH1(f)ρH1

(γ)

= lH1(f)ρH1

(z, w)

and glH1(f) ≤ lH1

(f).

Let z ∈ H1. By Theorem 4

ρH1(z, z + t)/|t| → ρH1

(z) as t → 0 and

therefore

ρH1(f(z), f(z + t))

|t|

=ρH1

(f(z), f(z + t))

|f(z) − f(z + t)||f(z) − f(z + t)|

|t|→ ρH1

(f(z))|f ′(z)|.

20

Now

glH1(f) ≥ ρH1

(f(z), f(z + t))

ρH1(z, z + t)

→ ρH1(f(z))|f ′(z)|ρH1

(z)

as t → 0 and lH1(f) ≤ glH1

(f). Since ρH1is

monotonic w.r.t the domain we have

glH1(f) ≤ 1. Proof for H2 is similar.

We say Hol(H1, H2) is Hj-strictly uniform if

lHj= sup

f∈Hol(H1,H2)

lHj(f) < 1.

For the inclusion map g : H2 → H1, H2 ⊂ H1

we denote the contraction constant

gl(H2, H1) = supz,w∈H2, z 6=w

ρH1(z, w)

ρH2(z, w)

and the infinitesimal contraction constant

l(H2, H1) = supz∈H2

ρH1(z)

ρH2(z)

.

21

Theorem 15. gl(H2, H1) = l(H2, H1) ≤ 1. If

H2 ( H1 then g is strict contraction and

infinitesimally strict contraction.

Proof. Proof of gl(H2, H1) = l(H2, H1) ≤ 1 is

similar to the proof of Theorem 14.

Let z, w ∈ H2 ( H1, z 6= w, and πj be universal

covering maps from B2 onto Hj with

πj(0) = z. By the proof of the Pick’s theorem g

lifts to a holomorphic map f from B2 to B2

such that f(0) = 0 and

π1 ◦ f = g ◦ π2. (16)

If ρH1(z) = ρH2

(z) then by taking derivatives

in (16) gives |f ′(0)| = 1. Schwarz lemma =⇒ f

is Möbius. This is contradiction, because f

cannot be surjective (∃p ∈ B2 with

π1(p) ∈ H1 \ H2 and p /∈ f(B2)). Therefore

l(H2, H1) < 1.

Similarly we can show that gl(H2, H1) < 1.

22

Corollary 17. If H2 is relatively compact

subdomain of H1, then l(H2, H1) < 1.

Proof. Follows from Theorems 7 and 15.

Definition 18. Subdomain D of hyperbolic

domain H is Lipschitz, if the inclusion map g

from D to H is infinitesimally strict

contraction.

Corollary 19. Every relatively compact

subdomain is Lipschitz.

23

We call

R(H2, H1) = supz∈H1, BH1

(z,r)⊂H2

r

the (hyperbolic) Bloch constant of H2, where

BH1(z, r) = {w ∈ H1 : ρH1

(z, w) < r}

for r > 0 and z ∈ H1.

Definition 20. Domain H2 ⊂ H1 is Bloch

subdomain if R(H2, H1) < ∞.

Theorem 21. [BCMN] Domain H2 ⊂ H1 is

Lipschitz iff H2 is Bloch subdomain.

24

References

[AST] V.V. Aseev, A.V. Sychëv, A.V. Tetenov:

Möbius-invariant metrics and generalized angles

in Ptolemaic spaces. (Russian) Sibirsk. Mat. Zh.

46 (2005), no. 2, 243–263; translation in Siberian

Math. J. 46 (2005), no. 2, 189–204.

[BCMN] A.F. Beardon, T.K. Carne, D. Minda, T.W.

Ng: Random iteration of analytic maps. Ergodic

Th. and Dyn. Systems 24 (2004), no. 3 659-675.

[B] D. Betsakos: Estimation of the hyperbolic

metric by using the punctured plane. Math. Z.

259 (2008), no. 1, 187–196.

[DD] M.-M. Deza, E. Deza: Dictionary of distances.

Elsevier, 2006.

[HMM] D. Herron, W. Ma, D. Minda: Möbius

invariant metrics bilipschitz equivalent to the

hyperbolic metric. Conform. Geom. Dyn. 12

(2008), 67–96.

[KL] L. Keen, N. Lakic: Hyperbolic geometry from a

local viewpoint. London Mathematical Society

Student Texts, 68. Cambridge University Press,

Cambridge, 2007.

[PT] A. Papadopoulos, M. Troyanov: Weak metrics

on Euclidean domains. JP J. Geom. Topol. 7

(2007), no. 1, 23–43.

25