Graphics Reference
In-Depth Information
()
()
=
(
()
)
zgt wgt fgt
=
,
=
,
(10.70)
1
2
1
where g
1
(t) and g
2
(t) have power series expansions in t that have a positive radius of
convergence. Such a variable t is called a (local)
uniformizing variable
and the process
is called
uniformization
. It exists because of the very definition of a Riemann surface
in terms of analytic elements and convergent power series. An expansion like in
(10.67) translates into a uniformization
01 2
2
zct wa atat
=+
,
= +
+
+
...,
with uniformizing variable t. For example, the square root in (10.65) is replaced by
an equation
12
w-=
0
(10.71)
and happens to have an especially simple uniformization
2
wt zt
=
,
=
.
(10.72)
The uniformization in (10.72) is actually valid for the whole Riemann surface asso-
ciated to (10.71) and not just to a neighborhood of 0. That is not always possible.
Whether or not a global uniformization exists for a Riemann surface is called the
uniformization problem
in complex analysis. The interested reader can find a nice
discussion of this problem and its history in [Abik81].
The pair of functions (g
1
(t),g
2
(t)) in equations (10.70) is just another representa-
tion of what we called an analytic element. As we just pointed out, from a topologi-
cal point of view it simply corresponds to showing that the Riemann surface is a
surface. Thinking of things in that way, since there are many ways to coordinatize a
neighborhood about a point in a manifold, why not try to find a special coordinate
neighborhood that gives us more information about this neighborhood. Define two
representations (g
1
(t),g
2
(t)) and (h
1
(t),h
2
(t)) to be
equivalent
if there exists a power
series
()
=+ +
2
ut
ct c t
... ,
c
π
0
,
(10.73)
1
2
1
which converges in a neighborhood of 0, and is one-to-one there, so that
()
=
(
()
)
ht gut
ht gut
,
.
1
1
()
=
(
()
)
2
2
One can prove that this is an equivalence relation and that if z has an expansion
m
m
+
1
za at
=+
+
a t
+
...
(10.74)
0
m
m
+
1
with m > 0 and a
m
π 0, then z has an equivalent expansion
m
za bs
=+
.
(10.75)
0
m
