the hyperbolic metric in complex analysisclaya/2015_hyperbolic...the hyperbolic distance between two...

The Hyperbolic Metric in Complex Analysis

Eric Schippers

October 15, 2015

Eric Schippers Hyperbolic metric October 15, 2015 1 / 41

Introduction

Geometric function theory

Geometric function theory is the study of geometric properties offamilies of complex analytic functions.

Examples:

Relation between the shape of the image domain and the analyticproperties of a function.Analytic properties which guarantee the functions are one-to-one.Moduli spaces of Riemann surfaces.Distribution of zeroes (value distribution theory).Hyperbolic geometry of analytic functions - special case of studyof “conformal metrics”.


Geometry of conformal metrics Conformal metrics and length

Definition of conformal metric

Domain = open, connected set in C.

Definition

A conformal metric on a domain D is a C2 function ρ : D → R+.

DefinitionThe ρ-length of a curve γ in D is

Lρ(γ) =

∫γρ(z)|dz|.

|dz| =

∣∣∣∣dzdt

∣∣∣∣dt =

√(dxdt

)2

+

(dydt

)2

dt

for z(t) = x(t) + iy(t)


Geometry of conformal metrics Conformal metrics and length

Terminology

A conformal metric is a special type of “Riemannian metric”.The term “metric” here is not the same as the term in analysis.However, every conformal metric gives rise to a distance function,which is in fact a metric.


Geometry of conformal metrics The hyperbolic metric on the disc

Hyperbolic metric

D = z : |z| < 1 ⊂ C

DefinitionThe hyperbolic metric on D is

λ(z) =1

1− |z|2.

The hyperbolic length of a curve γ in D is

L(γ) =

∫γ

|dz|1− |z|2

.

This is one example of a hyperbolic metric.



Isometries

DefinitionAn isometry is a one-to-one onto map f : D→ D such that for anycurve γ,

L(f γ) = L(γ).

Möbius transformations T : D D are isometries of the hyperbolicmetric:

T (z) = eiθ z − a1− az

⇒ |T ′(z)|1− |T (z)|2

=1

1− |z|2.

so

L(T γ) =

∫Tγ

|dw |1− |w |2

=

∫γ

|T ′(z)||dz|1− |T (z)|2

=

∫γ

|dz|1− |z|2

= L(γ).



Isometries continued

In fact these are all of them!

Wonderful coincidence:

Isometries =

T (z) = eiθ z − a

1− az: θ ∈ R, a ∈ D

= one-to-one, onto, analytic T : D→ D.



hyperbolic distance

DefinitionThe hyperbolic distance between two points z and w is

d(z,w) = infγ

∫γ

|dz|1− |z|2

.

DefinitionA geodesic segment between two points is a curve which attains theminimum distance.

Warning: This is not the usual definition, but in the case of thehyperbolic metric on the disc, it is equivalent to the usual one.



Geodesics and the distance function

Shortest path from 0 to z is the radial line segment:

0z

So the hyperbolic distance between 0 and z is:∫line

|dz|1− |z|2

=

∫ |z|0

dr1− r2

=12

log(

1 + |z|1− |z|

)= arctanh|z|.




So we can determine the shortest path through any points z and w : let

T (ζ) =ζ − w

1− ζw.

z w




So we can determine the shortest path through any points z and w : let

T (ζ) =ζ − w

1− ζw.

z w

0 = T (w)

T (z)T




So we can in turn determine the distance between z and w :

d(z,w) = d(T (z),T (w)) = d(T (z),0) = arctanh∣∣∣∣ z − w1− wz

∣∣∣∣=

12

log1 +

∣∣∣ z−w1−wz

∣∣∣1−

∣∣∣ z−w1−wz

∣∣∣ .



Hyperbolic world


Geometry of conformal metrics Curvature

Curvature

DefinitionLet D be a domain in C. Let ρ(z) : D → R+ be a conformal metric. Thecurvature of ρ is

K (z) = − 1ρ2(z)

∆ log ρ(z).

This is a special case of a more general notion in differential geometry.

What is curvature?

Theorem (Gauss-Bonnet theorem (special case))The sum of the interior angles of a triangle D is

π +

∫∫D

KdAρ.



Curvature

DefinitionLet D be a domain in C. Let ρ(z) : D → R+ be a conformal metric. Thecurvature of ρ is

K (z) = − 1ρ2(z)

∆ log ρ(z).

This is a special case of a more general notion in differential geometry.What is curvature?

Theorem (Gauss-Bonnet theorem (special case))The sum of the interior angles of a triangle D is

π +

∫∫D

KdAρ.



Example: geodesic triangles on the sphere

(x , y , z) : x2 + y2 + z2 = 1: geodesics are great circles, K = 1

sum of angles = 3π/2

π +∫∫

KdAρ = π + 1 · Area = π + π/2.



A bit more detail

I didn’t define curvature in enough generality to justify that last bit.You can use the definition I gave if you stereographically project:

(1) Trigonometry + work shows: the length of a curve on the sphere, isthe ρ-length of the projected curve if ρ(z) = 2/(1 + |z|2)(2) the ρ-area on the plane is the usual area on sphere(3) the curvature of ρ is 1.



A bit more detail


(1) Trigonometry + work shows: the length of a curve on the sphere, isthe ρ-length of the projected curve if ρ(z) = 2/(1 + |z|2)

(2) the ρ-area on the plane is the usual area on sphere(3) the curvature of ρ is 1.



A bit more detail


(1) Trigonometry + work shows: the length of a curve on the sphere, isthe ρ-length of the projected curve if ρ(z) = 2/(1 + |z|2)(2) the ρ-area on the plane is the usual area on sphere(3) the curvature of ρ is 1.



Hyperbolic case

If λ(z) = 1/(1− |z|2), then curvature is −4:

K (z) = − 4ρ2(z)

∂2

∂z∂zlog ρ

= 4(1− |z|2)2 ∂2

∂z∂zlog (1− zz)

= −4.



Hyperbolic triangles

Sum of angles < π.


Geometry of conformal metrics Pullbacks

Pull-back

DefinitionLet ρ be a metric on a domain Ω ⊂ C. Let f : D → Ω be an analyticmap such that f ′ 6= 0. The pull-back of ρ under f is

f ∗ρ(z) = ρ f (z) |f ′(z)|.

ρ

f ∗ρf

DΩ



Example

Let DR = z : |z| < R and

f : DR → Dz 7→ z/R

The pull-back of the hyperbolic metric λ(z) = 1/(1− |z|2) on D to DR

is

f ∗λ(z) = λ(f (z))|f ′(z)| =1/R

(1− |z/R|2)

=R

R2 − |z|2.



Idea of pull-back

Idea: the pull-back geometry on D is “the same” as the geometry on Ω.

Length is preserved: if γ is a curve in D

ρ-length(f γ) =

∫fγ

ρ(z)|dz| =

∫γρ(f (z))|f ′(z)||dz| = f ∗ρ-length(γ).

Curvature is preserved:

Kf∗ρ(z) = Kρ(f (z)).

Try it!


Geometry of conformal metrics The hyperbolic metric on an arbitrary domain

Completeness

DefinitionA conformal metric ρ is complete on a domain D if the associatedmetric space (D,dρ)is complete.

Completeness is preserved under pull-back: if f : D → Ω isone-to-one and onto, and ρ is a complete metric on Ω, then f ∗ρ is acomplete metric on D.

TheoremThe hyperbolic metric on D is complete.



Hyperbolic metric

DefinitionLet D be a domain in the plane. The hyperbolic metric of D is theunique complete metric on D with constant negative curvature −4(provided that it exists).

Example: The hyperbolic metric on D is λ(z) = 1/(1− |z|2).Example: The hyperbolic metric on DR is

λR(z) =R

R2 − |z|2.

Why? Because λR is the pull-back of the hyperbolic metric, and so it iscomplete and constant curvature −4.



Hyperbolic metric


Example: The hyperbolic metric on D is λ(z) = 1/(1− |z|2).

Example: The hyperbolic metric on DR is

λR(z) =R

R2 − |z|2.




Hyperbolic metric


Example: The hyperbolic metric on D is λ(z) = 1/(1− |z|2).Example: The hyperbolic metric on DR is

λR(z) =R

R2 − |z|2.



Geometry of conformal metrics The uniformization theorem

Uniformization theorem

Theorem (Uniformization theorem (Koebe, Poincaré))Every simply connected Riemann surface is biholomorphicallyequivalent to the Riemann sphere C, the complex plane C, or the unitdisk D.

Proof.Too much work for this talk.

Corollary

Every Riemann surface is given by the quotient of C, C or D by a nicegroup action (think tiling).



Almost everything is covered by D

• C only covers C• C only covers C, C\0 (cylinder) and tori.

• Everything else is D/G for some nice subgroup of the Möbiustransformations of the form


a ∈ D...

which are hyperbolic isometries. So almost every Riemann surfacehas a hyperbolic metric inherited from ∆.



Almost everything is covered by D

• C only covers C• C only covers C, C\0 (cylinder) and tori.• Everything else is D/G for some nice subgroup of the Möbiustransformations of the form


a ∈ D...

which are hyperbolic isometries. So almost every Riemann surfacehas a hyperbolic metric inherited from ∆.



Uniformization theorem

Nearly all domains have a hyperbolic metric.

Corollary (Uniformization theorem)Any subset of the plane which omits at least two points possesses ahyperbolic metric.


Complex analysis Introduction

Where’s the complex analysis?

The isometries of the hyperbolic metric are exactly the conformalautomorphisms of the disc.(Conformal automorphisms = one-to-one, onto analytic maps)

So there should be some connection between complex analysis on thedisc and hyperbolic geometry on the disc.Let’s look at some examples.


Complex analysis Ahlfors’ generalization of the Schwarz lemma

Schwarz lemma

TheoremIf f : D→ D is analytic and f (0) = 0 then |f (z)| ≤ |z|.

The “hyperbolically correct” version is

Theorem (hyperbolic Schwarz lemma)If f : D→ D is analytic then

d(f (z), f (w)) ≤ d(z,w).

Analytic maps from D to D are contractions.



proof of hyperbolic Schwarz lemma

LetT (z) =

z + w1 + wz

, S(ζ) =ζ − f (w)

1− f (w)ζ.

So if f : D→ D then S f T (0) = S(f (w)) = 0.

By the Schwarz lemma, |S(f (T (z)))| ≤ |z| ⇒ |S(f (z))| ≤ |T−1(z)| so∣∣∣∣∣ f (z)− f (w)

1− f (w)f (z)

∣∣∣∣∣ ≤∣∣∣∣ z − w1− wz

∣∣∣∣ .But arctanh is increasing so

d(f (z), f (w)) = arctanh

∣∣∣∣∣ f (z)− f (w)

1− f (w)f (z)

∣∣∣∣∣ ≤ arctanh∣∣∣∣ z − w1− wz

∣∣∣∣ = d(z,w).




LetT (z) =

z + w1 + wz

, S(ζ) =ζ − f (w)

1− f (w)ζ.

So if f : D→ D then S f T (0) = S(f (w)) = 0.By the Schwarz lemma, |S(f (T (z)))| ≤ |z| ⇒ |S(f (z))| ≤ |T−1(z)| so∣∣∣∣∣ f (z)− f (w)

1− f (w)f (z)

∣∣∣∣∣ ≤∣∣∣∣ z − w1− wz

∣∣∣∣ .

But arctanh is increasing so


∣∣∣∣∣ f (z)− f (w)

1− f (w)f (z)

∣∣∣∣∣ ≤ arctanh∣∣∣∣ z − w1− wz

∣∣∣∣ = d(z,w).




LetT (z) =

z + w1 + wz

, S(ζ) =ζ − f (w)

1− f (w)ζ.

So if f : D→ D then S f T (0) = S(f (w)) = 0.By the Schwarz lemma, |S(f (T (z)))| ≤ |z| ⇒ |S(f (z))| ≤ |T−1(z)| so∣∣∣∣∣ f (z)− f (w)

1− f (w)f (z)

∣∣∣∣∣ ≤∣∣∣∣ z − w1− wz

∣∣∣∣ .But arctanh is increasing so


∣∣∣∣∣ f (z)− f (w)

1− f (w)f (z)

∣∣∣∣∣ ≤ arctanh∣∣∣∣ z − w1− wz

∣∣∣∣ = d(z,w).



Ahlfors’ generalization of the Schwarz lemma

Theorem (Ahlfors-Schwarz lemma, special case)Let DR be the disc of radius R, with hyperbolic metric λR. For anymetric ρ on DR, such that the curvature Kρ(z) ≤ −4 for all z,

ρ(z) ≤ λR(z)

for all z.



The proof

Proof.For r < R we have Dr ⊂ DR. Let

v(z) =ρ

λr; z ∈ Dr .

v is continuous, positive, and v → 0 as |z| → r .

So v has a maximum; thus log v has a maximum; say at z0 ∈ Dr .

0 ≥ 4 log v(z0) = 4 log ρ−4 logλr

= −ρ2(z0)Kρ(z0) + λ2r (z0)Kλr (z0)

≥ 4ρ2(z0)− 4λ2r (z0).

So since z0 was the maximum, ρ(z) ≤ λr (z) for all z.Now let r → R.



The proof


v(z) =ρ

λr; z ∈ Dr .

v is continuous, positive, and v → 0 as |z| → r .So v has a maximum; thus log v has a maximum; say at z0 ∈ Dr .


= −ρ2(z0)Kρ(z0) + λ2r (z0)Kλr (z0)

≥ 4ρ2(z0)− 4λ2r (z0).




The proof


v(z) =ρ

λr; z ∈ Dr .



= −ρ2(z0)Kρ(z0) + λ2r (z0)Kλr (z0)

≥ 4ρ2(z0)− 4λ2r (z0).

So since z0 was the maximum, ρ(z) ≤ λr (z) for all z.

Now let r → R.



The proof


v(z) =ρ

λr; z ∈ Dr .



= −ρ2(z0)Kρ(z0) + λ2r (z0)Kλr (z0)

≥ 4ρ2(z0)− 4λ2r (z0).




Ahlfors

Ahlfors 1907-1996

According to Ahlfors: “This is an almost trivial fact and anyone whosees the need could prove it at once”.



Ahlfors continued

Finnish mathematician, advisors at University of Helsinki were E.Lindelöf and R. NevanlinnaFirst Fields Medal (with Jesse Douglas) in 1936 for work in valuedistribution theory (Nevanlinna theory).Wolf Prize in 1981Towering figure in Riemann surfaces and complex analysisMost famous as one of the founders of Teichmüller theory andquasiconformal mappings



Ahlfors’ generalization of the Schwarz lemma

Theorem (Ahlfors-Schwarz lemma, special case)Let DR be the disc of radius R, with hyperbolic metric λR. For anymetric ρ on DR, such that the curvature Kρ(z) ≤ −4 for all z,

ρ(z) ≤ λR(z)

for all z.

It says:The hyperbolic metric is maximal, among metrics with boundednegative curvatureIn particular, if f : DR → Ω, and λR is the hyperbolic metric on DR,and λΩ is the hyperbolic metric on Ω, then f ∗λΩ ≤ λR.



Schwarz lemma is special case

Let f : D→ D and λ(z) = 1/(1− |z|2) be the hyperbolic metric.

The curvature of f ∗λ equals −4 since curvature is invariant underpull-back.

So by the Ahlfors-Schwarz lemma

f ∗λ ≤ λ

so|f ′(z)|

1− |f (z)|2≤ 1

1− |z|2.


Complex analysis Applications of the Ahlfors-Schwarz lemma

Liouville’s theorem

TheoremLet f : C→ C be an analytic function such that |f (z)| ≤ M for all z ∈ C.Then f is constant.

Liouville’s theorem can be interpreted as a limiting case of Schwarzlemma.



Proof of Liouville’s theorem using the Schwarz lemma

Minda, Schober 1983.The hyperbolic metric on the disc of radius R is

λR(z) =R

R2 − |z|2.

For any R, f maps DR into DM .

By the Ahlfors-Schwarz lemma, for any R,

f ∗λM ≤ λR

soM|f ′(z)|

M2 − |f (z)|2≤ R

R2 − |z|2

for any fixed z.Letting R →∞, we get that |f ′(z)| = 0 for any z. So f = c.





λR(z) =R

R2 − |z|2.

For any R, f maps DR into DM .By the Ahlfors-Schwarz lemma, for any R,

f ∗λM ≤ λR

soM|f ′(z)|

M2 − |f (z)|2≤ R

R2 − |z|2

for any fixed z.

Letting R →∞, we get that |f ′(z)| = 0 for any z. So f = c.





λR(z) =R

R2 − |z|2.

For any R, f maps DR into DM .By the Ahlfors-Schwarz lemma, for any R,

f ∗λM ≤ λR

soM|f ′(z)|

M2 − |f (z)|2≤ R

R2 − |z|2

for any fixed z.Letting R →∞, we get that |f ′(z)| = 0 for any z. So f = c.



The Little Picard Theorem

Theorem (Little Picard Theorem)Let f : C→ C be an analytic mapping, whose image omits at least twopoints. Then f is constant.

The Little Picard theorem is really a case of the (Ahlfors-)Schwarzlemma in disguise.



proof of the Little Picard Theorem

Proof.Let p, q be the points omitted from the image of f . Let σ be thehyperbolic metric on C\p,q (uniformization theorem!)

For any R, f maps DR into C\p,q. Since f ∗σ has curvature −4, wemay apply the Ahlfors-Schwarz lemma:

σ(f (z))|f ′(z)| = f ∗σ ≤ λR(z) =R

R2 − |z|2.

Letting R →∞ (for any fixed z), we see that σ(f (z))|f ′(z)| = 0.But σ(f (z)) 6= 0, so |f ′(z)| = 0 (for any z)! So f is constant.

This approach due to Minda and Schober (1983). Actually this is avariation on the classical approach. They also give an elementaryproof without using the uniformization theorem.






σ(f (z))|f ′(z)| = f ∗σ ≤ λR(z) =R

R2 − |z|2.

Letting R →∞ (for any fixed z), we see that σ(f (z))|f ′(z)| = 0.

But σ(f (z)) 6= 0, so |f ′(z)| = 0 (for any z)! So f is constant.







σ(f (z))|f ′(z)| = f ∗σ ≤ λR(z) =R

R2 − |z|2.

Letting R →∞ (for any fixed z), we see that σ(f (z))|f ′(z)| = 0.But σ(f (z)) 6= 0, so |f ′(z)| = 0 (for any z)! So f is constant.



Conclusion

Summary of hyperbolic complex analysis theorems

The Schwarz lemma really says that analytic maps from D to Dare hyperbolic contractions.Liouville’s theorem is really a limiting case of the Schwarz lemma.The Little Picard Theorem is really a limiting case of the Schwarzlemma.Actually, any holomorphic map between hyperbolic Riemannsurfaces is a hyperbolic contraction.The hyperbolic metric is central to complex analysis.


References

Some References

1 D. Minda and G. Schober, Another elementary approach to thetheorems of Landau, Montel, Picard and Schottky. ComplexVariables 2 (1983) 157–164.

2 S. Krantz Complex analysis: the geometric viewpoint. CarusMathematical Monographs 23 (1990).

3 D. Kraus and O. Roth. Conformal metrics. arXiv:0805.2235v1(2008).

4 L. Ahlfors, An extension of Schwarz’ lemma. Transactions of theAmerican Mathematical Society 43 (1938) 359–364.

Upcoming book: D. Minda (Cincinnati) and A. Beardon (Cambridge),The hyperbolic metric in complex analysis. Springer.


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