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Klen 080908
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Transcript of Klen 080908
Analysis Seminar, Helsinki, 8.9.2008
Hyperbolic metric in
planar domains
Riku Klén
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.
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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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