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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.
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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)|.
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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.
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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.
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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.
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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.
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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)).
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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.
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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).
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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.
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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.
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• 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
) .
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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∗.
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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.
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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.
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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).
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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)|.
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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)
.
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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.
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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.
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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.
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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.
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