proof of the uniformization theorem

proof of the uniformization theorem

Our proof relies on the well-known

Newlander-Niremberg theorem

which implies, in particular, that any Riemmanian metric on an

oriented

2

-dimensional

real manifold

defines a unique

analytic structure

.

We will merely use the fact that

H

1

⁢

(

X

,

ℝ

)

=

0

. If

X

is

compact

, then

X

is a complex curve of genus

0

, so

X

≃

ℙ

1

. On the other hand, the elementary

Riemann mapping theorem

says that an open set

Ω

⊂

ℂ

with

H

1

⁢

(

Ω

,

ℝ

)

=

0

is either equal to

ℂ

or

biholomorphic

to the

unit disk

. Thus, all we have to show is that a non compact Riemann surface

X

with

H

1

⁢

(

X

,

ℝ

)

=

0

can be embedded in the

complex plane

ℂ

.

Let

Ω

ν

be an exhausting sequence of relatively compact

connected

open sets with smooth boundary in

X

. We may assume that

X

∖

Ω

ν

has no relatively compact

connected components

, otherwise we “fill the holes” of

Ω

ν

by taking the union with all such components. We let

Y

ν

be the double of the

manifold with boundary

(

Ω

¯

ν

,

∂

⁡

Ω

ν

)

, i.e. the union of two copies of

Ω

¯

ν

with opposite

orientations

and the boundaries identified. Then

Y

ν

is a compact oriented surface without boundary.

Fact: we have

H

1

⁢

(

Y

ν

,

ℝ

)

=

0

. We postpone the proof of this fact to the end of the present paragraph and we continue with the proof of the

uniformization theorem

.

Extend the almost

complex structure

of

Ω

¯

ν

in an arbitrary way to

Y

ν

, e.g. by an extension of a Riemmanian metric. Then

Y

ν

becomes a compact Riemann surface of genus

0

, thus

Y

ν

≃

ℙ

1

and we obtain in particular a

holomorphic

embedding

Φ

ν

:

Ω

ν

→

ℂ

. Fix a point

a

∈

Ω

0

and a non zero linear form

ξ

*

∈

T

a

⁢

X

. We can take the

composition

of

Φ

ν

with an affine linear map

ℂ

→

ℂ

so that

Φ

ν

⁢

(

a

)

=

0

and

d

⁢

Φ

ν

⁢

(

a

)

=

ξ

*

. By the well-known properties of injective holomorphic maps,

(

Φ

ν

)

is then uniformly bounded on every small disk centered at

a

, thus also on every compact subset of

X

by a connectedness argument. Hence

(

Φ

ν

)

has a subsequence converging towards an injective holomorphic map

Φ

:

X

→

ℂ

.

Proof of the ”fact”: Let us first compute the cohomology with compact support

H

c

1

⁢

(

Ω

ν

,

ℝ

)

. Let

u

be a closed

1

-form with compact support in

Ω

ν

. By Poincaré duality

H

c

1

⁢

(

X

,

ℝ

)

=

0

, so

u

=

d

⁢

f

for some ”test” function

f

∈

𝒟

⁢

(

X

)

. As

d

⁢

f

=

0

on a

neighborhood

of

X

∖

Ω

ν

and as all connected components of this set are non compact,

f

must be equal to the constant zero near

X

∖

Ω

ν

. Hence

u

=

d

⁢

f

is the zero class in

H

c

1

⁢

(

Ω

ν

,

ℝ

)

and we get

H

c

1

⁢

(

Ω

ν

,

ℝ

)

=

H

1

⁢

(

Ω

ν

,

ℝ

)

=

0

. The exact sequence of the pair

(

Ω

¯

ν

,

∂

⁡

Ω

n

⁢

u

)

yelds

ℝ

=

H

0

⁢

(

Ω

¯

ν

,

ℝ

)

→

H

0

⁢

(

∂

⁡

Ω

ν

,

ℝ

)

→

H

1

⁢

(

Ω

¯

ν

,

∂

⁡

Ω

ν

;

ℝ

)

≃

H

c

1

⁢

(

Ω

ν

,

ℝ

)

=

0

,

thus

H

0

⁢

(

∂

⁡

Ω

ν

,

ℝ

)

=

ℝ

. Finally, the

Mayer-Vietoris sequence

applied to small neighborhoods of the two copies of

Ω

¯

ν

in

Y

ν

gives an exact sequence

H

0

⁢

(

Ω

¯

ν

,

ℝ

)

⊕

2

→

H

0

⁢

(

∂

⁡

Ω

ν

,

ℝ

)

→

H

1

⁢

(

Y

ν

,

ℝ

)

→

H

1

⁢

(

Ω

¯

ν

,

ℝ

)

⊕

2

=

0

where the first map is onto. Hence

H

1

⁢

(

Y

ν

,

ℝ

)

=

0

.

References

J.-P. Demailly,

Complex Analytic

and

Algebraic Geometry

.

Title

proof of the uniformization theorem

Canonical name

ProofOfTheUniformizationTheorem

Date of creation

2013-03-22 15:37:50

Last modified on

2013-03-22 15:37:50

Owner

Simone (5904)

Last modified by

Simone (5904)

Numerical id

14

Author

Simone (5904)

Entry type

Proof

Classification

msc 30F20

Classification

msc 30F10

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