Download UNIFORMIZATION OF SURFACES COMPLEX ANALYSIS 8702 1. Riemann surfaces; Summary

UNIFORMIZATION OF SURFACES
COMPLEX ANALYSIS 8702
ALBERT MARDEN
1. Riemann surfaces; Summary
1. Let R denote a Riemann surface. This means that R has a complex structure, a
“rule” for measuring angles. More formally it means that {(Uα , fα )} is a system of open
neighborhoods covering R with associated homeomorphisms fα : Uα → f (Uα ) ⊂ C. The
key assumption is that on nonempty intersections
fα fβ−1 : fβ (Uα ∩ Uβ ) → fα (uα ∩ Uβ )
is a conformal map of each component. Via the maps {fα } and the transition property,
angles in C are carried up to well defined angles on R.
2. Denote by F = π1 (R) its fundamental group. As an abstract group, F is independent
of basepoint; the fundamental group at any basepoint O ∈ R is isomorphic (∼
=) to F .
3. Let H ⊂ F be a subgroup. For simplicity we will assume H is a normal subgroup.
This means that f Hf −1 = H; that is, f hf −1 = h0 ∈ H for all h ∈ H.
eH denote the covering surface of R corresponding to H. This means:
4. Let R
eH inherits its conformal structure from R and the locally injective projection
• R
eH → R which then becomes analytic.
π:R
eH ) ∼
• π1 (R
= H.
eH ) of cover transformations has the properties
• The group C(R
bH ).
(1) π(T z) = π(z), for all T ∈ C(R
∼
eH ) = F/H.
(2) C(R
The quotient group is well defined because H is a normal subgroup.
e of R is the case that H = {id}. In particular,
5. The universal covering surface R
e ∼
e is simply connected.
(1) π1 (R)
= {id}, that is, R
e ∼
e over
(2) C(R)
= F . That is every closed curve γ ∈ π1 (R, O) 6= id, and a point O∗ ∈ R
∗
e
O determines a unique cover transformation Tγ . The lift γ of γ to R from the
point O∗ terminates at Tγ (O∗ ) = O1∗ , a different point over O.
(3) The lift γ2∗ of γ from another point O2∗ over O determines a conjugate transformation
b is determined by O∗ and the element ρ ∈ π1 (R, O)
W = Sρ Tγ Sρ−1 . Here Sρ ∈ C(R)
with Sρ (O∗ ) = O2∗ . Then W (γ ∗ ) = γ2∗ .
Date: May 14, 2012.
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ALBERT MARDEN
2. Uniformization
Theorem A. Corresponding to each (abstract) simply connected Riemann surface there
is a conformal map onto exactly one of the following.
• S2 ,
• C = S2 \ {∞},
• D (unit disk or Möbius equivalent).
e is the universal cover of the Riemann surface R. For each
Theorem B. Suppose R
e
possiblity for R from Theorem A, there are the following possibilities for R. Each has its
own geometry as indicated:
e = S2 if and only if R = S2 . R then has spherical geometry.
• R
e = C if and only if
• R
(1) R = C,
(2) R = S2 \ {0, ∞},
(3) R = a torus.
In these cases the geometry of R is euclidean.
e = D for all other cases, and in all these cases the geometry of R is hyperbolic.
• R
e of
Corollary. Assume R 6= S2 , S2 \ {∞}, C. For each remaining case the group C(R)
cover transformations is generated as follows.
e = hz 7→ z + 1i,
• R = S2 \ {0, ∞}, C(R)
e
• R = a torus, C(R) = hz 7→ z + 1, z 7→ z + τ i,
• For all other types of R, R = D/Γ, where Γ is a fuchsian group. R is represented
by a fundamental hyperbolic polygon P ⊂ D for Γ; the orbit Γ(P ) of P tessellates
D.
e 6= S2 , under the
The term “uniformize” comes from the fact that the z coordinate from R
e → R, provides a “uniform” coordinate for R. By analogy, the projection
projection π : R
it
π(t) = e is a local homeomorphism of the real line R onto the unit circle. The unit circle
is thereby ‘uniformized’ by t ∈ R.