1Conformal transformations
III Schramm--Loewner Evolutions
1.1 Conformal transformations
Definition
(Conformal map)
.
Let
U, V
be domains in
C
. We say a holomorphic
function f : U → V is conformal if it is a bijection.
We will write
D
for the open unit disk, and
H
for the upper half plane. An
important theorem about conformal maps is the following:
Theorem
(Riemann mapping theorem)
.
Let
U
be a simply connected domain
with
U 6
=
C
and
z ∈ U
be any point. Then there exists a unique conformal
transformation f : D → U such that f(0) = z, and f
0
(0) is real and positive.
We shall not prove this theorem, as it is a standard result. An immediate
corollary is that any two simply connected domains that are distinct from
C
are
conformally equivalent.
Example.
Take
U
=
D
. Then for
z ∈ D
, the map promised by the Riemann
mapping theorem is
f(w) =
w + z
1 + ¯zw
.
In general, every conformal transformation f : D → D is of the form
f(w) = λ
w + z
1 + ¯zw
for some |λ| = 1 and z ∈ D.
Example. The map f : H → D given by
f(z) =
z − i
z + i
,
is a conformal transformation, with inverse
g(w) =
i(1 + w)
1 − w
.
In general, the conformal transformations H → H consist of maps of the form
f(z) =
az + b
cz + d
with a, b, c, d ∈ R and ad −bc 6= 0.
Example. For t ≥ 0, we let H
t
= H \ [0, 2
√
ti]. The map H
t
→ H given by
z 7→
p
z
2
+ 4t
is a conformal transformation. Observe that this map satisfies
|g
t
(z) −z| = |
p
z
2
+ 4t − z| → 0 as z → ∞.
So g
t
(z) ∼ z for large z.
Observe also that the family g
t
(z) satisfies the ODE
∂g
∂t
=
2
g
t
(z)
, g
0
(z) = z.
We can think of these functions
g
t
as being generated by the curve
γ
(
t
) = 2
√
ti
,
where for each
t
, the function
g
t
is the conformal transformation that sends
H \ γ([0, t]) to H (satisfying |g
t
(z) −z| → ∞).
Given the set of functions
g
t
, we can recover the curve
γ
as follows — for
each
z ∈ H
, we can ask what is the minimum
t
such that
g
t
is not defined on
z
. By ODE theorems, there is a solution up till the point when
g
t
(
z
) = 0, in
which case the denominator of the right hand side blows up. Call this time
τ
(
z
).
We then see that for each
t
, there is a unique
z
such that
τ
(
z
) =
t
, and we have
γ(t) = z.
More generally, suppose
γ
is any simple (i.e. non-self-intersecting) curve
in
H
starting from 0. Then for each
t
, we can let
g
t
be the unique conformal
transformation that maps
H\γ
([0
, t
]) to
H
with
|g
t
(
z
)
−z| → ∞
as above (we will
later see such a map exists). Then Loewner’s theorem says there is a continuous,
real-valued function W such that
∂g
t
∂t
=
2
g
t
(z) −W
t
, g
0
(z) = z.
This is the chordal Loewner equation. We can turn this around — given a
function
W
t
, what is the corresponding curve
γ
(
t
)? If
W
= 0, then
γ
(
t
) = 2
√
ti
.
More excitingly, if we take
W
t
=
√
κB
t
, where
B
t
is a standard Brownian motion,
and interpret this equation as a stochastic differential equation, we obtain
SLE
κ
.