THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS
OF QUADRATIC DIFFERENTIALS
JONATHAN TSAI
Abstract. We obtain a first order differential equation for the driving func-
tion of the chordal Loewner differential equation in the case where the domain
is slit by a curve which is a trajectory arc of certain quadratic differentials. In
particular this includes the case when the curve is a path on the square, trian-
gle or hexagonal lattice in the upper half-plane or, indeed, in any domain with
boundary on the lattice. We also demonstrate how we use this to calculate
the driving function numerically. Equivalent results for other variants of the
Loewner differential equation are also obtained: Multiple slits in the chordal
Loewner differential equation and the radial Loewner differential equation.
The proof of our theorem uses a generalization of Schwarz-Christoffel map-
ping to domains bounded by trajectory arcs of rotations of a given quadratic
differential.
Introduction
Suppose that H = {z ∈ C : Im(z) > 0} is the upper half-plane and γ : [0, T ) 7→ H
is a simple Jordan curve with γ(0) ∈ R and γ(0, T ) = {γ(t) : t ∈ (0, T )} ⊂ H. Then
for each t ∈ (0, T ),
Ht = H \ γ(0, t]
is a simply-connected domain and hence by the Riemann mapping theorem, we can
find a conformal map ft of H onto Ht. Moreover, we can require that ft has series
expansion
ft(z) = z −
as z → ∞.
Normalized in this way ft is unique and is said to be hydrodynamically normalized.
The function C(t) is positive, continuous and strictly increasing: it is called the half-
plane capacity of γ(0, t]. Thus we can reparameterize γ such that C(t) = 2t for all
t, we will call this parameterization by half-plane capacity. With this normalization
and parameterization, the function ft satisfies the differential equation (where f
denotes differentiation with respect to z and ḟt denotes differentiation with respect
to t):
(0.1) ḟt(z) = −
2f ′t(z)
z − ξ(t) ,
where ξ(t) = f−1t (γ(t)) is a continuous real-valued function. This is the chordal
Loewner differential equation; ξ(t) is called the driving function of the slit γ. The
converse is also true: given a measurable function ξ, the differential equation (0.1)
with initial condition f0(z) ≡ z has solution ft which is a conformal map from H
2000 Mathematics Subject Classification. Primary 30C20; Secondary 30C30, 60K35 .
http://arxiv.org/abs/0704.1933v3
2 J. TSAI
−4 −2 0 2 4 6 8
0 100 200 300 400 500 600
Figure 1. A path on the hexagonal lattice on the upper half-plane
(left) and a plot of its driving function on the y-axis against time
on the x-axis (right).
−2 0 2 4 6 8 10 12
0 50 100 150 200 250
Figure 2. A path on the square lattice on the upper half-plane
(left) and a plot of its driving function on the y-axis against time
on the x-axis (right).
into itself (although f(H) is not necessarily a slit domain). Chapter 3 and 4 of [7]
gives full details of this construction.
Since Schramm’s discovery of stochastic Loewner evolution in 1999 (see [15]),
there has been huge interest in the chordal Loewner differential equation and its
variants. But the relationship between the slit inH and its resulting driving function
is not well understood. There are a few papers that relate the behaviour of the slit
with the behaviour of the driving function e.g. [10],[8]; also, the paper [2] calculates
the slit arising from a few driving functions. In this paper, we will obtain a first
order differential equation for ξ (which we can then solve numerically) that allows
us to calculate the driving function ξ in the case where the curve γ is a trajectory
arc of a certain type of quadratic differential. We will show that this includes, for
example, the case when γ is a path on the square/triangle/hexagonal lattice in the
upper half-plane or indeed, in any domain whose boundary lies on such a lattice. So
for example, Figure 1 plots the driving function of a path on the hexagonal lattice
in the upper half-plane and Figure 2 plots the driving function of a path on the
square lattice in the upper half-plane.
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 3
We also note that we can obtain equivalent results for other variants of the
Loewner differential equation for example, in the radial version or with multiple
slits. We will discuss this in the paper as well.
The proof of our formulae uses a generalization of Schwarz-Christoffel mapping
to domains bounded by trajectory arcs of rotations of a given quadratic differential.
We also mention that, currently, the common method used to find the driving
function of a given slit is to use the Zipper algorithm discovered independently by
D. E. Marshall and R. Kühnau to approximate the function ft which can then be
used to determine the driving function. The Zipper algorithm can be viewed as
a discrete version of the Loewner differential equation and hence is well suited to
studying growth processes. It also has the advantage of being very fast. See [11]
and [3].
1. Main results
To state our main results, we have to provide some background in the theory of
quadratic differentials. Note that not all the terms used here are standard in the
literature. See Chapter 8 of [13] and [16] for more details. A quadratic differential
on a domain D ⊂ Ĉ = C ∪ {∞} is the formal expression
Q(z)dz2,
where Q(z) is a meromorphic function on D. Then for ω ∈ D with ω 6= ∞, Q(z)
has Laurent series expansion about ω,
Q(z) =
ak(z − ω)k
for some n > −∞ with an 6= 0. Then we define the degree of ω with respect to
Q(z)dz2, degQ(ω), to be equal to n.
If ∞ ∈ D, then near ∞, Q has Laurent series expansion given by
Q(z) =
then we define the degree of ∞ with respect to Q(z)dz2, degQ(∞) to be equal to
m − 4. The “4” in the definition ensures that the degree is conformally invariant
in a way which we will make precise later. Then ω ∈ D is:
• a zero of Q(z)dz2 if degQ(ω) > 0.
• a pole of Q(z)dz2 if degQ(ω) < 0.
• an ordinary point of Q(z)dz2 if degQ(ω) = 0.
A trajectory arc of Q(z)dz2 is a curve γ : (a, b) 7→ D that does not meet any
zeroes and poles of Q(z)dz2 and satisfies
Q(γ(t))γ̇(t)2 > 0 for all t ∈ (a, b).
For θ ∈ [0, π), a θ-trajectory arc of Q(z)dz2 is a curve γ : (a, b) 7→ D that satisfies
arg[Q(γ(t))γ̇(t)2] = 2θ for all t ∈ (a, b).
Then γ is a θ-trajectory arc of Q(z)dz2 if and only if it is a trajectory arc of
e−2iθQ(z)dz2. Hence, a 0-trajectory arc is simply a trajectory arc and we call a
π/2-trajectory arc an orthogonal trajectory arc. It is clear that these definitions
are invariant under reparameterization of γ so we will often call the point set of γ
4 J. TSAI
a trajectory arc or θ-trajectory arc. We call a maximal trajectory arc a trajectory
and similarly, a maximal θ-trajectory arc is called a θ-trajectory. For example, if we
consider the quadratic differential 1dz2 in C, then the θ-trajectories are the straight
lines with gradient exp(2θ).
We now consider a special type of quadratic differential: Let D be a domain
with piecewise analytic boundary. A Kühnau quadratic differential is a quadratic
differential, Q(z)dz2, on D satisfying the following two properties:
Definition: Let D be a domain with piecewise analytic boundary. A Kühnau qua-
dratic differential is a quadratic differential, Q(z)dz2, on D satisfying the following
two properties:
(1) We can write
such that each Γj is an open analytic arc with Γk ∩ Γj = ∅ for k 6= j
and moreover, Q(z) extends continuously to each Γj and arg[Q(z)dz
2] is
constant on each Γj i.e. each Γj is a θj-trajectory arc for some θj ∈ [0, π).
(2) At z ∈ Γk ∩ Γj for all j 6= k, there are either only finitely many direction
from which trajectories approach the point z or if there are infinitely many
directions from which trajectories approach the point z, then for each such
direction, there is only one trajectory that approaches z at this direction.
These quadratic differentials are studied by Kühnau in [6] where he applies them
to the study of certain Grötzsch-style extremal problems
Property (i) above, also implies that D is locally connected. Thus each prime
end of D corresponds to a unique point in ∂D (see [14, p. 27]). If, in addition,
a point on ∂D corresponds to a unique prime end, then we make no distinction
between the two. Let z be a prime end of D. Then we have 2 cases: Either z ∈ Γj
for some j = 1, . . . , n; or there exist exactly 2 of the (Γj), that end at the prime end
z. In the latter case, we will denote z by zk and assume that Γk, a θk-trajectory
arc, and Γk−1, a θk−1-trajectory arc, are the only 2 arcs that end at zk. Then we
can define the degree of zk in D with respect to Q(z)dz
2, degD,Q(zk), as follows:
degD,Q(zk) =
2[|θk − θk−1|/π + Jk − 1] if θk 6= θk−1,
2Jk if θk = θk−1,
where Jk is the number of trajectories of Q(z)dz
2 inside D that end at the prime
end zk. If Jk is infinite, then the degree is not defined. Then for prime ends z such
that z ∈ Γj for some j = 1, . . . , n, we define
degD,Q(z) = 0.
Although the motivation for this definition currently seems unclear, we will see
that this indeed generalizes the concept of degree to points on the boundary. In
particular, we will show that for x ∈ ∂H, if deg
H,Q(x) ∈ Z, then Q can be extended
to a meromorphic function in a neighbourhood of x with
H,Q(x) = degQ(x).
We then have the following theorem on Kühnau quadratic differentials in H:
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 5
Theorem 1.1. Suppose that Q(z)dz2 is a Kühnau quadratic differential on H.
Then we have
Q(z) = R
(z − ζj)λj
for some constant R 6= 0, ζj ∈ C, λj ∈ R for j = 1, . . . , n.
This theorem can be viewed as a generalization of the Schwarz-Christoffel for-
mula to domains bounded by θk-trajectory arcs of a given quadratic differential.
We then have the following theorem on the Loewner driving function of a φ-
trajectory arc of a Kühnau quadratic differential Q(z)dz2 that starts at a point
ξ0 ∈ R with degH,Q(ξ0) = N ∈ {0, 1, . . .}.
Theorem 1.2. Suppose that Q(z)dz2 is a Kühnau quadratic differential on H such
that there is a point ξ0 ∈ R with degH,Q(ξ0) = N ∈ {0, 1, . . .}; then we have
Q(w) = (w − ξ0)N
(w − aj)αj
where aj ∈ C and αj ∈ R. Let γ : [0, T ) 7→ ∂H be a simple curve such that
γ(0) = ξ0, γ(0, T ) ⊂ H and γ(0, T ) is a φ-trajectory arc of Q(z)dz2 (φ ∈ [0, π))
that is parameterized by half-plane capacity. Suppose that the functions ft maps
H conformally onto H \ γ(0, t] and are hydrodynamically normalized. Then for
t ∈ (0, T )
(1.1) 2ξ(t) = −µ−C−(t)− µ+C+(t)−
αjAj(t)
+Σ0,
(1.2) ξ̇(t) = − µ
C−(t)− ξ(t) −
C+(t)− ξ(t) −
Aj(t)− ξ(t)
with initial condition ξ(0) = ξ0. Where the functions Aj(t) are defined by
Aj(t) = f
t (aj) for j = 1, . . . , n,
and C+(t) > C−(t) are the two preimages of ξ0 under ft;
µ± = deg
H\γ(0,t],Q(ft(C
±(t))),
Nξ0 +
We can then use Theorem 1.2 to find the driving function in the case when the
slit γ consists of consecutive θk-trajectory arcs of given quadratic differentials. We
will explain how to do this in further detail later. One difficulty with using Theorem
1.2 is that the parameterization is inherently given in terms of half-plane capacity.
This makes it difficult to calculate the driving function ξ if we do not know anything
about the half-plane capacity of the trajectory arc (which, in general, is the case).
The next theorem will allow us to compare the parametrization with the length of
the slit:
6 J. TSAI
Theorem 1.3. Suppose that Q(z)dz2, γ and ft are as defined in Theorem 1.2. Let
Φt(z) =
Q(ft(z))f
(z − ξ(t))2 .
Then γ satisfies
(1.3) γ̇(t) = −2
Φt(ξ(t))
Q(γ(t))
The rest of this paper is organized as follows: In the Section 2, we will state
some basic results from the theory of quadratic differentials and use them to prove
Theorem 1.1. Then we will use Theorem 1.1 to prove Theorems 1.2 and 1.3 in Sec-
tion 3. In Section 4 we will discuss how to obtain the driving function numerically
using Theorems 1.2 and 1.3. Finally in Section 5, we will discuss extensions of The-
orem 1.2 to the case with multiple slits as well as to the radial Loewner differential
equation.
2. Kühnau quadratic differentials and generalized
Schwarz-Christoffel mapping
The aim of this section is to prove Theorem 1.1. We will first look at some of
the basic results in the theory quadratic differentials that we will need.
Transformation Law.
Suppose that f is a conformal map from a domain D2 onto a domain D1 and
suppose that Q1(w)dw
2 is a quadratic differential on D1. If we define
(2.1) Q2(z) ≡ Q1(f(z))f ′(z)2
then Q2(z)dz
2 is a quadratic differential on D2. Then, it is clear that θ-trajectory
arcs are preserved by this transformation law i.e.
γ is a θ-trajectory arc of Q2(z)dz
2 ⇔ f ◦ γ is a θ-trajectory arc of Q2(w)dw2 ,
and also, for z ∈ D2
degQ2(z) = degQ1(f(z)).
Hence trajectories and degQ are conformally invariant in the above sense.
The following lemma tells us that the behaviour of a quadratic differential at a
neighbourhood of a point is determined by the degree of that point.
Lemma 2.1 (Local behaviour of quadratic differentials). Let Q(z)dz2 be a qua-
dratic differential on a domain D. Then for every ω ∈ D there is a conformal
mapping w = φ(z) of some neighbourhood of ω such that
Q(z)dz2 =


dw2 if degQ(ω) = 0,
wndw2 if degQ(ω) = n ≥ 1,
w−ndw2 if degQ(ω) = −n ≤ −1 with n odd,
c2w−2dw2 if degQ(ω) = −2,
(w−n + cw−1)2dw2 if degQ(ω) = −n ≤ −4 with n even.
Here, c is the residue of a branch of
Q(z) at ω.
Proof. See Theorem 8.1 of [13] or Section 6 of [16]. �
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 7
So since trajectories are conformally invariant this lemma tells us that the local
structure of trajectories around a point ω ∈ D is completely determined by degQ(ω)
and the converse is true as well.
Lemma 2.2. Suppose that ω ∈ D and degQ(ω) = n. Then
(1) For n ≥ −1, there are exactly n + 2 trajectories of Q(z)dz2 that end at ω
and form equal angles with each other.
(2) For n ≤ −3, there are infinitely many trajectories ending at ω and more-
over, there are |n| − 2 directions at ω forming equal angles such that the
trajectories approach ω in these directions.
(3) For n = −2, the behaviour depends on the value of c (as defined in Lemma
2.1).
(a) If c is real, then the trajectories are the images of all radial lines under
the map φ defined in Lemma 2.1.
(b) If c is purely imaginary, then the trajectories are the images of all
concentric circles under the map φ defined in Lemma 2.1.
(c) If Re[c], Im[c] 6= 0 then the trajectories are the images of logarithmic
spirals under the map φ defined in Lemma 2.1.
Proof. See Section 7 of [16]. �
This lemma shows that it makes sense for us to define degD,Q(x), the degree of
a point on the boundary, in terms of the trajectories ending at x. If D = H and Q
extends to a meromorphic function on a neighbourhood of some x ∈ R∪ {∞} with
H,Q(x) finite. Then by studying the trajectory structure at x, we can see that
H,Q(x) = degQ(x).
This is the motivation for defining degD,Q in the way we have. The next lemma
shows that the degD,Q is also conformally invariant:
Lemma 2.3. Suppose that Q(z)dz2 is a Kühnau quadratic differential on a domain
D and f is a conformal map of the upper half-plane H onto D. Then the quadratic
differential Qf (w)dw
2 on H, defined by Qf(w) = Q(f(w))f
′(w)2, is also a Kühnau
quadratic differential. Moreover, suppose that z ∈ ∂D is a prime end of D. Then
degD,Q(z) = degH,Qf (f
−1(z)).
Proof. By Carathéodory’s theorem, f extends continuously to ∂H and by Schwarz’s
reflection, f extends analytically across f−1(Γk) for all k = 1, . . . , n. Since θ-
trajectory arcs are conformally invariant, this implies that Qf defined by (2.1) is
a Kühnau quadratic differential on H. Moreover, for all k = 1, . . . , n, f(Γk) is a
θk-trajectory arc of Qf (z)dz
2. Also each point on ∂H corresponds bijectively to a
prime end of H. Hence there is a bijective correspondence between the points of
∂H and prime ends of D. Then
degD,Q(z) = degH,Qf (f
−1(z))
follows from the conformal invariance of trajectories. �
Reflection across trajectories.
Suppose that D is a domain such that Γ ⊂ ∂D is an open interval in R. Let
Q(z)dz2 be a quadratic differential such that Γ is a trajectory arc or an orthogonal
8 J. TSAI
trajectory arc of Q(z)dz2. Then let D− = {z : z ∈ D} be the reflection of D along
Γ. Define
Q−(z) = Q(z) for z ∈ D−.
Then since Γ is a trajectory, we have
Q(z) = Q−(z) ∈ R for z ∈ Γ.
Thus by defining
(2.2) Q∗(z) =
Q(z) for z ∈ D,
Q−(z) for z ∈ D−,
Q(z) = Q−(z) for z ∈ Γ.
it is easy to see that Q∗ is meromorphic in D ∪ D− and hence Q∗(z)dz2 is a
quadratic differential on D ∪ D−. Thus by the transformation law (and using
Schwarz reflection), this shows that we can extend quadratic differentials across
trajectory arcs or orthogonal trajectory arcs.
We will use reflection to prove the following lemma:
Lemma 2.4. Suppose Q(z)dz2 is a Kühnau quadratic differential on H. Then for
any z ∈ ∂H, deg
H,Q(z) ∈ Z implies that Q(z)dz2 extends to a quadratic differential
on a neighbourhood of z and hence
H,Q(z) = degQ(z).
Proof. Firstly, if z ∈ Γj for some j = 1, . . . , n. Then by definition degH,Q(z) = 0
and Q(z)dz2 can be extended to a neighbourhood of z by reflection. By definition,
every z ∈ Γj is an ordinary point of Q(z)dz2 and hence degQ(z) = degH,Q(z) = 0.
Otherwise we write z = zk and suppose that a θk−1-trajectory arc, Γk−1, and a
θk-trajectory arc, Γk, end at zk. Then, by definition, degH,Q(zk) ∈ Z implies that
θk − θk−1 is a multiple of π/2. Thus Γk−1 and Γk are trajectory arcs or orthogonal
trajectory arcs of e−2iθk−1Q(z)dz2. Thus by reflection, e−2iθk−1Q(z)dz2 extends to
a neighbourhood of zk. Hence, Q(z)dz
2 also extends to a neighbourhood of zk. �
We can now prove Theorem 1.1; but first, we explain briefly why we can view
Theorem 1.1 as a generalized form of Schwarz-Christoffelmapping: Schwarz-Christoffel
mapping is a method of computing the conformal map between the upper half-plane
and a domain bounded by a polygon. See [12] for more details. If we have a confor-
mal map f fromH to some domainD such that the sides ofD consist of θ-trajectory
arcs of the quadratic differential Q(w)dw2. Then Q(w)dw2 is a Kühnau quadratic
differential on D and hence by Lemma 2.3, Q(f(z))f ′(z)dz2 is a Kühnau quadratic
differential on H. Theorem 1.1 then implies that
Q(f(z))f ′(z)2 = R
(z − ζj)λj
This is precisely the Schwarz-Christoffel formula when Q(z) ≡ 1.
Also, we comment that the case when Q(w)dw2 is either negative or positive on
R (i.e. the boundary of R consists only of trajectory arcs and orthogonal trajectory
arcs) is easy to prove: we can use reflection to extend Q(z)dz2 to a quadratic
differential on the Riemann sphere Ĉ. Hence Q(z) must be rational since property
(ii) in the definition of Kühnau quadratic differentials guarantees that Q(z) does not
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 9
have any essential singularities and so Q(z) is rational (since the only meromorphic
functions on Ĉ are rational). This proves Theorem 1.1 for this case.
Proof of Theorem 1.1. Since Q(z)dz2 is a Kühnau quadratic differential, we can
z1 < . . . < zm,
(zk−1, zk) for k = 1, . . . ,m,
(zm,∞) for k = m+ 1,
(−∞, z0) for k = 0,
such that each Γk is a θk-trajectory arc of Q(z)dz
2 for some θk ∈ [0, π). Let
T = {Γ1, . . . ,Γm+1} .
Then take any Γ ∈ T . Since Γ is a θ-trajectory for some θ, Γ is a trajectory
arc of e−2iθQ(z)dz2; hence by reflection, we can reflect the quadratic differential
e−2iθQ(z)dz2 across Γ to get a quadratic differential on H− = {Im(z) < 0} which
we call Q̃(z)dz2. Similarly, by rotating Q̃(z)dz2, we can reflect it across another
Υ ∈ T to get another quadratic differential Q∗(z)dz2 on H. Since Q∗ is obtained
from Q by rotating twice, we have
Q∗(z) = eiσQ(z)
for some σ ∈ [0, 2π). This shows that
Ψ(z) =
Q′(z)
(Q∗)′(z)
Q∗(z)
can be extended to a meromorphic function in C\{z1, . . . , zm}. Then part (ii) of the
definition of Kühnau quadratic differentials implies that all the finite singularities of
Ψ(z) are simple poles otherwise Q(z) would have an essential singularity which, by
the great Picard theorem, contradicts part (ii) of the definition of Kühnau quadratic
differentials. Thus we can write:
Ψ(z) = h(z)−
z − ζj
where ζj ∈ C, and λj ∈ R, and h(z) is an entire function in C that does not does
not vanish in C. This implies that
Q(z) = exp
h(ζ)dζ
(z − ζj)λj
(z − zk)νk
Moreover, the singularity at ∞ of
h(ζ)dζ
cannot be essential by part (ii) of the definition of Kühnau quadratic differentials
(otherwise we would get a contradiction with the great Picard theorem as above).
This implies that
h(ζ)dζ
10 J. TSAI
is constant (since it has no zeroes or poles). Hence
Q(z) = R
(z − ζj)λj
If ζj ∈ C \ R, then by definition, we must have
λj = degQ(ζj).
Moreover, if ζj ∈ R and degH,Q(ζj) <∞ we also have
νk = degH,Q(zk).
We will not prove this fact here but in the following corollary we will consider a
special case. The general proof follows readily from it. We will prove the following
corollary which is simply an application of Theorem 1.1 to domains slit by φ-
trajectory arcs:
Corollary 2.5. Suppose that Q(z)dz2 is a Kühnau quadratic differential on H such
that there is a point ξ0 ∈ R with degH,Q(ξ0) = N ∈ {0, 1, . . .}; then we can write
(2.3) Q(w) = R(w − ξ0)N
(w − aj)αj
where aj ∈ C, αj ∈ R, and also R is some non-zero constant. Let γ : [0, T ] 7→ H be a
simple curve such that γ(0) = ξ0 and γ(0, T ) is a φ-trajectory arc of Q(w)dw
2 in H
(φ ∈ [0, π)) and ζ = γ(T ) ∈ H is an ordinary point of Q(w)dw2 (i.e. degQ(ζ) = 0).
Suppose that f maps H conformally onto H \ γ(0, T ]. Then f satisfies
(2.4) Q(f(z))f ′(z)2 = R′(z − ξ)2(z − c−)µ
(z − c+)µ
(z −Aj)αj ,
where R′ is some constant; c−, c+ are the two preimages of ξ0 under f satisfying
c− < c+; Aj is the preimage of aj under f ; and ξ is the preimage of ζ; and
µ± = deg
H\γ(0,s],Q(f(c
Proof. Theorem 1.1 and Lemma 2.4 imply that Q(w) can be written as (2.3). Then
Lemma 2.3 implies that Q̂(z) = Q(f(z))f ′(z)2 is a Kühnau quadratic differential.
So by Theorem 1.1, we only need to look at the singularities of Q̂(z)dz2.
Now, by Schwarz reflection, f extends to a conformal map on C \ {ξ, c−, c+}.
Thus, by the conformal invariance of trajectories, this implies that
Qf (z) = R
′(z − ξ)M (z − c−)µ
(z − c+)µ
(z −Aj)αj .
Then by Lemma 2.2, there are exactly two φ-trajectory arcs of Q(z)dz2 ending
at ζ = γ(T ) of which γ(0, T ) is one of them. So by the conformal invariance of
trajectories, there is one φ-trajectory arcs of Q̂(z)dz2 ending at ξ that is contained
in H. Hence, by definition, deg bQ,H(ξ) = 2. Using Lemma 2.4, this implies that
degQ(ξ) = 2 i.e. M = 2. Thus we only need to determine µ
− and µ+.
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 11
Note that since ξ0 has degree N with respect to Q(z)dz
2, we can determine,
using Lemma 2.2, that the angle between γ(0, T ) and f((ξ, c−)) at ξ0 is
πψ− = π
degH,Q(f(c
−)) + 2
N + 2
and similarly, the angle between γ(0, T ) and f((ξ, c−)) at ξ0 is
πψ+ = π
degH,Q(f(c
+)) + 2
N + 2
Hence, by Schwarz reflection, the function
F (z) = (f(z)− ξ0)1/ψ
extends to a conformal mapping on a neighbourhood of c−. Thus in a neighbour-
hood of z = c−, we can write
(2.5) f(z) = ξ0 + (z − c−)ψ
h(z)ψ
where h is analytic in a neighbourhood of z = c− with h(c−) 6= 0. Now
Q′f (z)
Qf (z)
Q′(f(z))f ′(z)2
Q(f(z))
f ′′(z)
f ′(z)
The residue at z = c− of the left-hand side of the equation is µ−, and we can use
(2.5) to determine the residue at z = c− of the right-hand side. Thus we get
µ− = degH,Q(f(c
We apply the same method to c+ to get µ+. �
3. Domains slit by θ-trajectory arcs
Let Q(w)dw2 be a Kühnau quadratic differential on H with deg
H,Q(ξ0) = N ∈
{0, 1, . . .} for some ξ0 ∈ R. Then by Theorem 1.1 and Lemma 2.4,
Q(w) = (w − ξ0)N
(w − aj)αj
where αj ∈ R and aj ∈ C. Now suppose that γ : [0, T ) 7→ H is a simple curve
such that γ(0) = ξ0 and γ(0, T ) is a φ-trajectory of Q(z)dz
2 in H (φ = [0, π))
that is parameterized by half-plane capacity. As mentioned in the introduction,
there exists conformal maps ft : H 7→ Ht = H \ γ(0, t] satisfying the hydrodynamic
normalization. Then by restricting Q(w)dw2 to a quadratic differential on Ht we
can induce via ft and (2.1), a quadratic differential on H:
(3.1) Qt(z)dz
2 = Q(ft(z))f
2dz2.
We now use Corollary 2.5 and (3.1) to prove Theorem 1.2.
Proof of Theorem 1.2. Note that by Schwarz reflection, each ft can be extended
to a conformal map on Ĉ \ {C−(t), C+(t), ξ(t)}. Then since ft(z) satisfies the
hydrodynamic normalization, this implies that
f ′t(z)
2 = 1 +O
as z → ∞.
12 J. TSAI
So by (3.1),
(3.2)
Qt(z)
Q(ft(z))
= 1 +O
as z → ∞.
If we let ζ = 1/z, then we get
(3.3)
Qt(1/ζ)
Q(ft(1/ζ))
= 1 +O
as ζ → 0.
Since ft is analytic in a neighbourhood of infinity, (3.3) is a Taylor series expansion
and hence we can look at the Taylor series coefficients, in particular:
C(0,ǫ)
f ′(1/ζ)2
C(0,ǫ)
Qt(1/ζ)
ζ2Q(ft(1/ζ))
dζ = 0
for small enough ǫ > 0 where C(0, ǫ) is the anticlockwise contour about the circle
with centre at zero and radius ǫ > 0. Then by Theorem 1.1 and Corollary 2.5, we
can write
Q(w) = R(w − ξ0)N
(w − aj)αj
Qt(z) = R
′(z − ξ)2(z − C−(t))µ
(z − C+(t))µ
(z −Aj(t))αj .
Hence by the residue theorem (since ft(1/ζ) = 1/ζ + · · · as ζ → 0), this implies
2ξ(t) + µ
(t) + µ
(t) +
αkAk(t)
Nξ0 +
This implies (1.1). To get (1.2), note that ft satisfies the chordal Loewner differen-
tial equation (0.1) and hence if we let Gt = f
t ◦ fs for some s ∈ (0, T ) fixed and
t > s, then the chain rule implies that Gt satisfies the differential equation
(z) =
Gt(z)− ξ(t)
Then for some s sufficiently close to t, we can write each Aj(t) = Gt(wj) wj ∈ C
for all j = 1, . . . , n. Thus
Ȧj(t) =
Aj(t)− ξ(t)
Similarly, we get
Ċ±(t) =
C±(t)− ξ(t) .
Hence we get (1.2) from differentiating (1.1). �
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 13
An extension:
We can extend Theorem 1.2 to the case when γ is made up of different θk-trajectory
arcs of some quadratic differential Q(z)dz2: Let γ : (0, T ] 7→ H be a curve with
γ(0) ∈ R such that there is a partition
{0 = t0 < t1 < · · · < tr = T }
such that γ(tk−1, tk) is a θk-trajectory arc of Q(z)dz
2 and γ(tk) is an ordinary
point of Q(z)dz2 for k = 1, . . . , r. Then we can find the driving function ξ(t) of γ
by applying Theorem 1.2 to the θ1-trajectory arc γ(0, t1) to get a driving function
ξ1(t), and applying Theorem 1.2 inductively to each f
(γ(tk, tk+1)) (which is a
θk+1-trajectory arc of the quadratic differential Qtk(z)dz
2 = Q(ftk(z))f
(z)2dz2)
to get ξk(t). Then
ξ(t) = ξk(t) for t ∈ [tk−1, tk).
We also have the following corollary:
Corollary 3.1. Suppose that Q(w)dw2 and γ are as defined in Theorem 1.2.
Then the driving function ξ and Aj , C
−, C+ as defined in Theorem 1.2 are in
C∞(0, T ). Moreover, we can write any derivative of ξ, C−, C+, Aj explicitly in
terms of ξ, C−, C+, Aj and the exponents µ
−, µ+, αj.
Proof. Recall that, in the proof of Theorem 1.2, we had the formulae
Ȧj(t) =
Aj(t)− ξ(t)
, Ċ±(t) =
C±(t)− ξ(t) .
This implies that each term in (1.2) is differentiable so we can write the second
derivative of ξ in terms of ξ(t), Aj(t), C
±(t). This in turn implies that we can write
the third derivative of ξ in terms of ξ(t), Aj(t), C
±(t) and the exponents. Continuing
inductively, we have showed that every derivative of ξ exists and can be expressed
in terms of ξ(t), Aj(t), C
±(t) and the exponents. Note that each derivative of ξ is
finite for t ∈ (0, T ) since
|Aj(t)− ξ(t)|, |C±(t)− ξ(t)| > 0.
Then ξ is smooth implies that Aj(t), C
±(t) are also smooth. �
Theorem 1.3 then follows from Corollary 2.5 and Theorem 1.2:
Proof of Theorem 1.3.
First note that, by the definition of θ-trajectory arcs, γ̇ always exists and is never
0. Also by Corollary 2.5, Φt(ξ(t)) 6= 0,∞; thus the right hand side of (1.3) always
exists since, by definition, γ avoids poles and zeroes of Q(w)dw2.
Recall that ft(ξ(t)) = γ(t), this implies that
γ̇(t) = ḟt(ξ(t)) + f
t(ξ(t))ξ̇(t).
Then combining the Loewner differential equation (0.1) with (3.1) we have
ḟt(z) = −
z − ξ(t)
Qt(z)
Q(ft(z))
Φt(z)
Q(ft(z))
⇒ ḟt(ξ(t)) = −2
Φt(ξ(t))
Q(γ(t))
Note that Φt(ξ(t)) 6= 0,∞ since, by Corollary 2.5, Qt(z) has a double zero at ξ(t).
14 J. TSAI
Thus we have
γ̇(t) = −2
Φt(ξ(t))
Q(γ(t))
Qt(ξ(t))
Q(γ(t))
ξ̇(t) = −2
Φt(ξ(t))
Q(γ(t))
since Theorem 1.2 implies that ξ̇ is finite for all t ∈ (0, T ) and Corollary 2.5 implies
that Qt(ξ(t)) = 0. �
4. Applying Theorem 1.2
In practice, understanding ξ(t) via (1.1) is not possible: it is difficult to calculate
the positions of the zeroes and poles of Qt because the information we have on them
is all relative to ξ(t) (which we are trying to find). On the other hand, (1.2) is more
useful in applications. In this section, we will demonstrate how we can use (1.2)
to calculate numerically the driving function of a given slit that consists of θk-
trajectory arcs of a given quadratic differential. The method is basically a modified
version of Euler’s method.
Firstly, for any smooth function h on (0,T), Taylor’s theorem implies that for all
M = 1, 2, . . .,
(4.1)∣∣∣∣∣h
h(t) +
)∣∣∣∣∣ ≤
M !KM
s∈(t,t+ 1K )
for t, t + 1/K ∈ (0, T ). We will apply (4.1) to the functions ξ, Ak and C± (as
defined in Theorem 1.2) noting that, by Corollary 3.1, they are smooth and all of
their derivatives can be expressed in terms of ξ(t), Ak(t) and C
±(t). Thus if we
know ξ(s), Ak(s) and C
±(s) we can use (4.1) to obtain an approximate formula for
ξ(s+K−1), Ak(s+K
−1) and C±(s+K−1) (choosing K to be small and/or M to
be large so that the right-hand-side of (4.1) is small); then we can apply (4.1) to
ξ(s+K−1), Ak(s+K
−1) and C±(s+K−1) to find ξ(s+2K−1), Ak(s+2K
−1) and
C±(s+2K−1). Continuing like this, we obtain an approximation of ξ at the points
{s+ nK−1}.
So clearly what we need to do now is find the starting values ξ(s), Ak(s) and
C±(s) so we can apply the above method. But because ξ is not differentiable at
0, we cannot use the formula (4.1) with t = 0. The way around this is to note
that if deg
H,Q(ξ0) = N ∈ {0, 1, 2, . . .} then since we know degH,Qt(C
+(t)), we can
calculate the angle that the trajectory makes with the line [ξ0,∞) (as in the proof
of Corollary 2.5). Then we find that the angle is πψ where:
πψ = π
2 deg
(C+(t)) + 2
N + 2
So if we choose s small enough, we have
fs ≈ Fψ,ξ0s ,
where Fψ,ξ0s is the conformal map that maps H conformally onto H
s that is hydro-
dynamically normalized where Hψs is the upper half-plane slit by the straight line
starting at ξ0 making an angle πψ with [ξ0,∞), with half-plane capacity 2s. Then
we also have
Ak(s) ≈ (Fψ
−1(ak),
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 15
and also, C−(s), C+(s) are approximately the two preimages of ξ0 under F
ψ+,ξ0
Then we can use (1.1) to calculate ξ(s) approximately. We can then plug this
information into (4.1) as described above.
Note that F
t can be found using the fact that
(4.2) F
λt (z) =
z − (1− 2p)
z − (1− 2p)
t+ (1− p)
for some λ. Then we reparameterize this formula to remove the λ and translate the
point 0 to ξ0. Unfortunately, inverting this function cannot be done explicitly but
it can be done numerically very efficiently using Newton’s method. Alternatively,
by selecting a small s, we can assume that
Ak(s) ≈ ak
for all k. Then we note that the 2 preimages of ξ0 under F
s can be determined
explicitly (see [11]). This obviates the need to numerically invert F
Another difficulty is that, in general, given a slit, we cannot parameterize it by
half-plane capacity so it would be difficult, for example, to know at which t one
should stop. Most formulae for calculating half-plane capacity of some compact set
K rely on knowing the conformal map fK of H onto H\K (normalized hydrodynam-
ically). One possibility would be to use the probabilistic definitions of half-plane
capacity given in [7]. We will use the fact that Theorem 1.3 and Corollary 3.1 imply
that we can give all derivatives of γ(t) in terms of ξ(t), Ak(t), C
−(t), C+(t) and the
exponents µ−, µ+, αk so if we know these, we can also use (4.1) to approximate
γ. This in turn allows us to calculate the length of the slit γ. Thus if we know
beforehand length of our slit, we can calculate at what value of t we stop.
We now have everything we need in order to use (4.1) to calculate the driving
function numerically of any slit that is made up of θk trajectory arcs of a qua-
dratic differential Q(w)dw2. We will demonstrate how this is done in the following
example:
An example. Suppose that γ : (0, T ) → H is a piecewise linear arc parameterized
by half-plane capacity that satisfies:
• γ(0) = 0.
• From t = 0 to t = t1, γ is the straight line arc from 0 to i; call this Γ1.
• From t = t1 to t = t2, γ is the straight line arc from i to 2 + i; call this Γ2.
• From t = t2 to t = t3 = T , γ is the straight line arc from 2 + i to 2 + 2i;
call this Γ3.
First note that γ is made up of alternating (π/2)- and 0-trajectory arcs of the qua-
dratic differential 1dw2 in H and hence we can use Theorem 1.2 (or more specifically
the extension of Theorem 1.2 detailed in Section 3) to calculate ξ̇. As mentioned
previously, there is no easy way to know beforehand what t1, . . . , t3 are. For sim-
plicity, we will only use M = 1 in (4.1) i.e.
≈ f(t) + ḟ(t)
and fix a large K. Obviously Γ1 forms a right angle with real line; so we can use
(4.2) to determine the function
ft1 = F
1/2,0
(z) =
z2 − 4t1.
16 J. TSAI
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
0 1 2 3 4 5 6
Figure 3. The example path in the upper half-plane (left) and a
plot of its driving function on the y-axis against time on the x-axis
(right).
It is easy to see that in this case, t1 = 1/4 and ξ is constantly 0 for t ∈ (0, t1].
This induces the quadratic differential using (3.1):
Qt1(z)dz
z2dz2
(z + 1)(z − 1) .
Hence, we let A1(t1) = −1, A2(t1) = 1. Also f−1t1 (γ2) is a 0-trajectory arc of
Q1(z)dz
2 starting from ξ(t1) = 0 on R (by the conformal invariance of trajectories).
Now note that f−1t1 (γ2) makes an angle of π/4 with the positive real axis. and hence
ft1+K−1 ≈ F
1/4,0
sinceK is large. We can then use Newton’s method to find the preimages under the
above approximation of ft1+K−1 of the points A1(t1), A2(t1) and the 2 preimages of
zero to get the points A1(t1+K
−1), A2(t1+K
−1), C−(t1+K
−1), C+(t1+K
−1) and
hence, using (1.1), we can find ξ(t1+K
−1). Then inserting this into (4.1), as detailed
above we can also find ξ(t1 + nK
−1) and A1(t1 + nK
−1), A2(t1 + nK
−1), C−(t1 +
nK−1), C+(t1 + nK
−1); also, by Theorem 1.3, we can find |γ̇(t1 + nK−1)| if we let
t2(K) = inf
|γ̇(t1 + nK−1)| > (length of Γ2) = 2
then t2(K) ≈ t2 for K large. So we just assume that t2 = t2(K). Let A3(t2) =
C−(t2) and A4(t2) = C
+(t2). Hence by (3.1),
Qt2(z)dz
(z − ξ(t2))2(z −A3(t2))dz2
(z −A1(t2))(z −A2(t2))(z −A4(t2))
Then, by the conformal invariance of trajectories, f−1t2 (Γ3) is a π/2-trajectory of
Qt2(z)dz
2 and also, f−1t2 (Γ3), makes an angle 3π/4 with (ξ(t2),∞) and so
ft2+K−1 ≈ F
3/4,0
Then, as before, we can use Newton’s method to find the preimages under the above
approximation of ft2+K−1 of the points A1(t2), . . . , A4(t2) and the 2 preimages of
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 17
ξ(t2) to get the points A1(t2+K
−1), . . . , A4(t2+K
−1), C−(t2+K
−1), C+(t2+K
and hence use (1.1) to get ξ(t2+K
−1). We insert these into the formula iteratively
to get ξ(t2+nK
−1) and A1(t2+nK
−1), . . . , A4(t2+nK
−1), C−(t2+nK
−1), C+(t2+
nK−1) until t2+nK
−1 ≈ T . Thus the end result is that we found the driving func-
tion of the first 3 steps of the slit given in Figure 3. Of course, our calculation of ξ
will be more accurate by taking larger K.
For example, we can use the above method to calculate the driving function of
any path on the square/triangle/hexagonal lattice on H starting from some point in
R. In fact we can calculate the driving function of a path on the square/triangle/hexagonal
lattice in any polygon D by mapping the half-plane conformally onto D and pulling
back the quadratic differential 1dw2 on D to Q(z)dz2 on H using the transforma-
tion law. Also note that, in general, any curve γ can be approximated by a curve
γδ which lies on the square lattice δZ
2. Then it can be shown that
ξδ → ξ uniformly as δ ց 0,
where ξδ is the driving function of γδ and ξ is the driving function of γ hence, we
can use the above method to calculate ξδ then take the limit as δ ց 0 to obtain ξ.
Another point to note is that using the above method, we do not need to know
before hand what the trajectory arc of the given quadratic differential looks like;
so for arbitrary Kühnau quadratic differentials, we can use this method to plot the
trajectories starting at the boundary.
We end this section by looking at what happens when the slit approaches the
boundary:
Proposition 4.1. Suppose that γ : [0, T ) 7→ H is a simple curve such that γ(0) ∈ R
and γ(0, T ) is a θ-trajectory arc of some quadratic differential Q(z)dz2. Then let ξ
be the driving function of γ. If
γ(t) ∈ R ∪ γ(0, T ),
i.e. γ makes a loop at time T . Then
∣∣∣∣→ ∞ as tր T
for all n = 0, 1, . . ..
Proof.
For t ∈ (0, T ), we define
Γ(t) = {γ(s) : s ∈ (t, T )}.
Then Γt is a θ-trajectory arc in Ht = H\γ(0, t] of Q(w)dw2 and it is also a crosscut
in Ht (see [14]). Then by the conformal invariance of θ-trajectories, f
t (Γt) ⊂ H
is a θ-trajectory arc of Qt(z)dz
2. Moreover, f−1t (Γt) is a crosscut of H with one
end point at ξ(t) and the other end point in R such that either C+(t) or C−(t)
is contained in the closure of the bounded component of H \ f−1t (Γt). Without
loss of generality, assume it is C+(t). Then since diam(f−1t (Γt)) → 0 as t ր T ,
we must have ξ(t) = C+(T ) and hence by (1.2), ξ̇(t) → ∞ as t ր T . Similarly,
we differentiate (1.2) as mentioned in Corollary 3.1 to obtain the result for higher
order derivatives. �
18 J. TSAI
This means that as γ gets closer and closer to making a loop, the approximation
by (4.1) stops working no matter what M we choose. This phenomenon can be
observed in Figure 3, as we turn the last corner in γ, we can see that ξ decreases
faster even though the slit is not yet that close to the boundary.
5. Generalizing Theorem 1.2
5.1. Multiple slits. Suppose that γk : [0, T ) → H for k = 1, . . . , N are disjoint
simple curves such that γk(0) ∈ R and γk(0, T ) ⊂ H. By the Riemann mapping
theorem, there exists unique ft that map H conformally onto Ht = H\
k=1 γk(0, t]
that satisfies the hydrodynamic normalization. We can reparameterize such that
γk(0, T ]
has half-plane capacity 2t. Then ft satisfies
(5.1) ḟt(z) = −2f ′t(z)
bk(t)
z − ξk(t)
where
bk(t) = 1,
and ξk(t) = f
t (γk(t)). See [2] for more details.
Theorem 5.1. Suppose that Q(w)dw2 is a Kühnau quadratic differential on H
such that the points ξk(0) ∈ R satisfy
H,Q(ξk(0)) = βk ∈ {0, 1, 2, . . .}
for all k. Then we can write
Q(w) =
(w − ξk(0))βk
(w − aj)αj
with aj ∈ C and αj ∈ R. Then suppose that γk : [0, T ) → H for k = 1, . . . , N are
disjoint simple curves such that γk(0) ∈ R and γk(0, T ) ⊂ H and are parameterized
as above. Then
(5.2) 2
ξk = −
(µ−k C
k (t) + µ
k (t))
αjAj(t)
+Σ0,
(5.3)
ξ̇l(t) =
k=1,k 6=l
bk(t)
ξl(t)− ξk(t)
bl(t)
(t)− ξl(t)
bl(t)
(t)− ξl(t)
αjbl(t)
Aj(t)− ξl(t)
for all l ∈ {1, . . . , N}. Where C−k (t) and C
k (t) are the two preimages of ξk(0)
under ft satisfying C
k (t) < C
k (t);
µ±k = degHt,Q(ft(C
k (t)));
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 19
Aj(t) = f
t (aj); and
βkξk(0)
Proof. By Theorem 1.1 and Lemma 2.4, we can write
Q(w) =
(w − ξk(0))βk
(w − aj)αj
Then either by modifying the proof of Corollary 2.5 or iterating N slit functions
and applying Corollary 2.5 N times, it is not too difficult to see that if we define
Qt(z) by (3.1), then
(5.4)
Qt(z) =
(z − ξk(t))2(z − C−k (t))
k (z − C+k (t))
(z −Aj(t))αj
Then the proof of (5.2) is exactly the same as the proof of (1.1) in Theorem 1.2.
The proof of (5.3) is more complicated. First let
Pt(z) = −2
bk(t)
z − ξk(t)
Then (5.1) becomes
ḟt(z) = f
t(z)Pt(z).
Now take the logarithmic derivative of Qt(z) with respect to z and t separately
using the definition of Qt(z) given by (3.1) to get
Q′t(z)
Qt(z)
Q′(ft(z))f
Q(ft(z))
f ′′t (z)
f ′t(z)
Q̇t(z)
Qt(z)
Q′(ft(z))ḟt(z)
Q(ft(z))
ḟ ′t(z)
f ′t(z)
Q′(ft(z))f
t(z)Pt(z)
Q(ft(z))
f ′′t (z)Pt(z) + f
t(z)P
t (z)
f ′t(z)
where we substitute (5.1) in for ḟt to get from the first to the second line. Thus we
(5.5)
Q̇t(z)
Qt(z)
Q′t(z)
Qt(z)
Pt(z) + 2P
t(z).
So then by (5.4), we note that
Q′t(z)
Qt(z)
z − ξk(t)
z − C−k (t)
z − C+k (t)
z −Aj(t)
Q̇t(z)
Qt(z)
2ξ̇k(t)
z − ξk(t)
µ−k Ċ
k (t)
z − C−k (t)
µ+k Ċ
k (t)
z − C+k (t)
αjȦj(t)
z −Aj(t)
20 J. TSAI
0 2 4 6 8 10
Figure 4. A plot of the two driving functions ξ1 (top) and ξ2
(bottom) on the y-axis against time on the x-axis.
Thus substituting this into (5.5) and comparing the coefficient of
z − ξl(t)
(i.e. the residue at z = ξl(t) of both sides of (5.5)), we find that this is exactly
(5.3). �
Similarly, we can prove a version of Theorem 1.3 and Corollary 3.1 for multiple
slits. This means that we can use the method detailed in Section 4 with (5.3)
to calculate the driving function for multiple θk-trajectory arc slits. For example
Figure 4 plots the graph of the two driving functions ξ1 and ξ2 in the case when
γ1 and γ2 are 2 vertical slits starting from -1 and 1 (i.e. orthogonal trajectories of
1dz2) and growing at the same speed. Compare this with Figure 7 in [2].
5.2. Radial Loewner evolution. The chordal Loewner differential equation was
introduced because the upper half-plane was an easier domain to work with for
many applications but the original setting of the Loewner differential equation is in
the unit disc D = {z ∈ C : |z| < 1}: Suppose that γ : [0, T ) 7→ D is a simple curve
such that γ(0) ∈ T = {z : |z| = 1} and γ(0, T ) ⊂ D \ {0}. Then Dt = D \ γ(0, t]
is simply-connected and 0 ∈ Dt for all t ∈ (0, T ). Hence the Riemann mapping
theorem implies that there is unique conformal map ft mapping D conformally onto
Dt such that ft(0) = 0 and f
t(0) > 0. Then Schwarz’s lemma and the Carathéodory
kernel theorem implies that f ′t(0). is strictly decreasing and continuous so we can
reparameterize such that f ′t(0) = e
−t. f ′t(0) is sometimes called the conformal
radius of Dt; hence in this case we are parameterizing by conformal radius. Then
the functions ft satisfy the radial Loewner differential equation:
ḟt(z) = −zf ′t(z)
z + eiξ(t)
z − eiξ(t)
See [9] for more details.
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 21
Theorem 5.2. Suppose that Q(w)dw2 is a Kühnau quadratic differential on D such
that degQ(0) = K ∈ Z and eiξ0 ∈ T = {|z| = 1} satisfies
= N ∈ {0, 1, 2, . . .}
then we have
Q(w) = wK(w − eiξ0)N
(w − aj)αj
where aj ∈ C and αj ∈ R. Then if γ : [0, T ) 7→ D is a simple curve such that
γ(0) = eiξ0 and γ(0, T ) ⊂ D \ {0} is a φ-trajectory arc of Q(w)dw2 in D that does
not meet 0 and is parameterized as above. Then we have
(5.6) e2iξ(t) = e−2tΠ0C
−(t)−µ
C+(t)−µ
Aj(t)
−αj ,
(5.7)
ξ̇(t) = − 1
C−(t) + eiξ(t)
C−(t)− eiξ(t)
C+(t) + eiξ(t)
C+(t)− eiξ(t)
Aj(t) + e
iξ(t)
Aj(t)− eiξ(t)
where, as usual, the functions Aj(t) are defined by
Aj(t) = f
t (aj) for j = 1, . . . , n,
µ± = degDt,Q(ft(C
±(t))),
C+(t) > C−(t) are the two preimages of eiξ0 under ft; and also,
Π0 = e
Proof. The formula for Q(w) can be obtained from Theorem 1.1 by the transforma-
tion law. We then define Qt by (3.1). Since the point 0 is fixed by ft, this implies
that the degQt(0) = degQ(0) = K. Thus we can apply Corollary 2.5 (again, using
the transformation law) to get
Qt(z) = z
K(z − eiξ(t))2(z − C−(t))µ
(z − C+(t))µ
(z −Aj(t))αj
Then since by definition,
Qt(z) = Q(ft(z))f
This immediately implies (5.6) by substituting z = 0. Then we get (5.7) in the
same way as we get (1.2) from (1.1) in the proof of Theorem 1.2. �
As in the case of multiple slits, a version of Theorem 1.3 and Corollary 3.1 holds
for this case.
22 J. TSAI
5.3. Other versions of the Loewner differential equation. There are several
other versions of the Loewner differential equation for simply-connected domains in
the literature; the methods in this paper should work in those cases as well and the
proofs should be similar to the proofs of Theorem 1.2 etc. Also, [4], [5] generalizes
the Loewner differential equation to multiply-connected domains and again, some
of the methods should work in these cases possibly using methods in [1] to extend
Theorem 1.1 to multiply-connected domains. Finally, even if we consider general
2-dimensional growth processes given by the Loewner-Kufarev differential equation
(see Chapter 6 of [13]), some of the methods in this paper should still be applicable.
Acknowledgement. The author would like to express his gratitude to his super-
visor Dr. T. K. Carne for his constant guidance and helpful discussion when writing
this paper.
References
1. D. Crowdy. The Schwarz-Christoffel mapping to bounded multiply connected polygonal do-
mains. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461(2061):2653–2678, 2005.
2. W. Kager, B. Nienhuis, and L. P. Kadanoff. Exact solutions for Loewner evolutions. J. Statist.
Phys., 115(3-4):805–822, 2004.
3. T. Kennedy. Computing the Loewner driving process of random curves in the half plane.
arXiv:math/0702071v1
4. Y. Komatu. Untersuchungen über konforme Abbildung von zweifach zusammenhängenden
Gebieten. Proc. Phys.-Math. Soc. Japan (3), 25:1–42, 1943.
5. Y. Komatu. On conformal slit mapping of multiply-connected domains. Proc. Japan Acad.,
26(7):26–31, 1950.
6. R. Kühnau. Über die analytische Darstellung von Abbildungsfunktionen insbesondere von
Extremalfunktionen der Theorie der konformen Abbildung. J. Reine Angew. Math., 228:93–
132, 1967.
7. G. F. Lawler. Conformally invariant processes in the plane. American Mathematical Society,
USA, 2005.
8. J. R. Lind. A sharp condition for the Loewner equation to generate slits. Ann. Acad. Sci.
Fenn. Math., 30(1):143–158, 2005.
9. K. Löwner. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I. Math.
Ann., 89, 1923.
10. D. E. Marshall and S. Rohde. The Loewner differential equation and slit mappings. J. Amer.
Math. Soc., 18(4):763–778 (electronic), 2005.
11. D. E. Marshall and S. Rohde. Convergence of the zipper algorithm for conformal mapping.
preprint, 2006.
12. Z. Nehari. Conformal mapping. Dover Publications, New York, 1982.
13. C. Pommerenke. Univalent functions. Vandenhoeck & Ruprecht, Göttingen, 1975. With
a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische
Lehrbücher, Band XXV.
14. C. Pommerenke. Boundary behaviour of conformal maps. Springer-Verlag, Berlin, 1992.
15. O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel
J. Math., 118:221–288, 2000.
16. K. Strebel. Quadratic differentials. Springer-Verlag, Berlin, 1984.
Department of Mathematics, Chinese University of Hong Kong, Shatin, New Terri-
tories, Hong Kong
E-mail address: jtsai@math.cuhk.edu.hk
http://arxiv.org/abs/math/0702071
	Introduction
	1. Main results
	Definition:
	2. Kühnau quadratic differentials and generalized Schwarz-Christoffel mapping
	Transformation Law
	Reflection across trajectories
	3. Domains slit by -trajectory arcs
	An extension:
	4. Applying Theorem ??
	An example.
	5. Generalizing Theorem ??
	5.1. Multiple slits
	5.2. Radial Loewner evolution
	5.3. Other versions of the Loewner differential equation
	Acknowledgement.
	References

  We obtain a first order differential equation for the driving function of the
chordal Loewner differential equation in the case where the domain is slit by a
curve which is a trajectory arc of certain quadratic differentials. In
particular this includes the case when the curve is a path on the square,
triangle or hexagonal lattice in the upper halfplane or, indeed, in any domain
with boundary on the lattice. We also demonstrate how we use this to calculate
the driving function numerically. Equivalent results for other variants of the
Loewner differential equation are also obtained: Multiple slits in the chordal
Loewner differential equation and the radial Loewner differential equation. The
method also works for other versions of the Loewner differential equation. The
proof of our formula uses a generalization of Schwarz-Christoffel mapping to
domains bounded by trajectory arcs of rotations of a given quadratic
differential that is of interest in its own right.

Introduction
Suppose that H = {z ∈ C : Im(z) > 0} is the upper half-plane and γ : [0, T ) 7→ H
is a simple Jordan curve with γ(0) ∈ R and γ(0, T ) = {γ(t) : t ∈ (0, T )} ⊂ H. Then
for each t ∈ (0, T ),
Ht = H \ γ(0, t]
is a simply-connected domain and hence by the Riemann mapping theorem, we can
find a conformal map ft of H onto Ht. Moreover, we can require that ft has series
expansion
ft(z) = z −
as z → ∞.
Normalized in this way ft is unique and is said to be hydrodynamically normalized.
The function C(t) is positive, continuous and strictly increasing: it is called the half-
plane capacity of γ(0, t]. Thus we can reparameterize γ such that C(t) = 2t for all
t, we will call this parameterization by half-plane capacity. With this normalization
and parameterization, the function ft satisfies the differential equation (where f
denotes differentiation with respect to z and ḟt denotes differentiation with respect
to t):
(0.1) ḟt(z) = −
2f ′t(z)
z − ξ(t) ,
where ξ(t) = f−1t (γ(t)) is a continuous real-valued function. This is the chordal
Loewner differential equation; ξ(t) is called the driving function of the slit γ. The
converse is also true: given a measurable function ξ, the differential equation (0.1)
with initial condition f0(z) ≡ z has solution ft which is a conformal map from H
2000 Mathematics Subject Classification. Primary 30C20; Secondary 30C30, 60K35 .
http://arxiv.org/abs/0704.1933v3
2 J. TSAI
−4 −2 0 2 4 6 8
0 100 200 300 400 500 600
Figure 1. A path on the hexagonal lattice on the upper half-plane
(left) and a plot of its driving function on the y-axis against time
on the x-axis (right).
−2 0 2 4 6 8 10 12
0 50 100 150 200 250
Figure 2. A path on the square lattice on the upper half-plane
(left) and a plot of its driving function on the y-axis against time
on the x-axis (right).
into itself (although f(H) is not necessarily a slit domain). Chapter 3 and 4 of [7]
gives full details of this construction.
Since Schramm’s discovery of stochastic Loewner evolution in 1999 (see [15]),
there has been huge interest in the chordal Loewner differential equation and its
variants. But the relationship between the slit inH and its resulting driving function
is not well understood. There are a few papers that relate the behaviour of the slit
with the behaviour of the driving function e.g. [10],[8]; also, the paper [2] calculates
the slit arising from a few driving functions. In this paper, we will obtain a first
order differential equation for ξ (which we can then solve numerically) that allows
us to calculate the driving function ξ in the case where the curve γ is a trajectory
arc of a certain type of quadratic differential. We will show that this includes, for
example, the case when γ is a path on the square/triangle/hexagonal lattice in the
upper half-plane or indeed, in any domain whose boundary lies on such a lattice. So
for example, Figure 1 plots the driving function of a path on the hexagonal lattice
in the upper half-plane and Figure 2 plots the driving function of a path on the
square lattice in the upper half-plane.
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 3
We also note that we can obtain equivalent results for other variants of the
Loewner differential equation for example, in the radial version or with multiple
slits. We will discuss this in the paper as well.
The proof of our formulae uses a generalization of Schwarz-Christoffel mapping
to domains bounded by trajectory arcs of rotations of a given quadratic differential.
We also mention that, currently, the common method used to find the driving
function of a given slit is to use the Zipper algorithm discovered independently by
D. E. Marshall and R. Kühnau to approximate the function ft which can then be
used to determine the driving function. The Zipper algorithm can be viewed as
a discrete version of the Loewner differential equation and hence is well suited to
studying growth processes. It also has the advantage of being very fast. See [11]
and [3].
1. Main results
To state our main results, we have to provide some background in the theory of
quadratic differentials. Note that not all the terms used here are standard in the
literature. See Chapter 8 of [13] and [16] for more details. A quadratic differential
on a domain D ⊂ Ĉ = C ∪ {∞} is the formal expression
Q(z)dz2,
where Q(z) is a meromorphic function on D. Then for ω ∈ D with ω 6= ∞, Q(z)
has Laurent series expansion about ω,
Q(z) =
ak(z − ω)k
for some n > −∞ with an 6= 0. Then we define the degree of ω with respect to
Q(z)dz2, degQ(ω), to be equal to n.
If ∞ ∈ D, then near ∞, Q has Laurent series expansion given by
Q(z) =
then we define the degree of ∞ with respect to Q(z)dz2, degQ(∞) to be equal to
m − 4. The “4” in the definition ensures that the degree is conformally invariant
in a way which we will make precise later. Then ω ∈ D is:
• a zero of Q(z)dz2 if degQ(ω) > 0.
• a pole of Q(z)dz2 if degQ(ω) < 0.
• an ordinary point of Q(z)dz2 if degQ(ω) = 0.
A trajectory arc of Q(z)dz2 is a curve γ : (a, b) 7→ D that does not meet any
zeroes and poles of Q(z)dz2 and satisfies
Q(γ(t))γ̇(t)2 > 0 for all t ∈ (a, b).
For θ ∈ [0, π), a θ-trajectory arc of Q(z)dz2 is a curve γ : (a, b) 7→ D that satisfies
arg[Q(γ(t))γ̇(t)2] = 2θ for all t ∈ (a, b).
Then γ is a θ-trajectory arc of Q(z)dz2 if and only if it is a trajectory arc of
e−2iθQ(z)dz2. Hence, a 0-trajectory arc is simply a trajectory arc and we call a
π/2-trajectory arc an orthogonal trajectory arc. It is clear that these definitions
are invariant under reparameterization of γ so we will often call the point set of γ
4 J. TSAI
a trajectory arc or θ-trajectory arc. We call a maximal trajectory arc a trajectory
and similarly, a maximal θ-trajectory arc is called a θ-trajectory. For example, if we
consider the quadratic differential 1dz2 in C, then the θ-trajectories are the straight
lines with gradient exp(2θ).
We now consider a special type of quadratic differential: Let D be a domain
with piecewise analytic boundary. A Kühnau quadratic differential is a quadratic
differential, Q(z)dz2, on D satisfying the following two properties:
Definition: Let D be a domain with piecewise analytic boundary. A Kühnau qua-
dratic differential is a quadratic differential, Q(z)dz2, on D satisfying the following
two properties:
(1) We can write
such that each Γj is an open analytic arc with Γk ∩ Γj = ∅ for k 6= j
and moreover, Q(z) extends continuously to each Γj and arg[Q(z)dz
2] is
constant on each Γj i.e. each Γj is a θj-trajectory arc for some θj ∈ [0, π).
(2) At z ∈ Γk ∩ Γj for all j 6= k, there are either only finitely many direction
from which trajectories approach the point z or if there are infinitely many
directions from which trajectories approach the point z, then for each such
direction, there is only one trajectory that approaches z at this direction.
These quadratic differentials are studied by Kühnau in [6] where he applies them
to the study of certain Grötzsch-style extremal problems
Property (i) above, also implies that D is locally connected. Thus each prime
end of D corresponds to a unique point in ∂D (see [14, p. 27]). If, in addition,
a point on ∂D corresponds to a unique prime end, then we make no distinction
between the two. Let z be a prime end of D. Then we have 2 cases: Either z ∈ Γj
for some j = 1, . . . , n; or there exist exactly 2 of the (Γj), that end at the prime end
z. In the latter case, we will denote z by zk and assume that Γk, a θk-trajectory
arc, and Γk−1, a θk−1-trajectory arc, are the only 2 arcs that end at zk. Then we
can define the degree of zk in D with respect to Q(z)dz
2, degD,Q(zk), as follows:
degD,Q(zk) =
2[|θk − θk−1|/π + Jk − 1] if θk 6= θk−1,
2Jk if θk = θk−1,
where Jk is the number of trajectories of Q(z)dz
2 inside D that end at the prime
end zk. If Jk is infinite, then the degree is not defined. Then for prime ends z such
that z ∈ Γj for some j = 1, . . . , n, we define
degD,Q(z) = 0.
Although the motivation for this definition currently seems unclear, we will see
that this indeed generalizes the concept of degree to points on the boundary. In
particular, we will show that for x ∈ ∂H, if deg
H,Q(x) ∈ Z, then Q can be extended
to a meromorphic function in a neighbourhood of x with
H,Q(x) = degQ(x).
We then have the following theorem on Kühnau quadratic differentials in H:
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 5
Theorem 1.1. Suppose that Q(z)dz2 is a Kühnau quadratic differential on H.
Then we have
Q(z) = R
(z − ζj)λj
for some constant R 6= 0, ζj ∈ C, λj ∈ R for j = 1, . . . , n.
This theorem can be viewed as a generalization of the Schwarz-Christoffel for-
mula to domains bounded by θk-trajectory arcs of a given quadratic differential.
We then have the following theorem on the Loewner driving function of a φ-
trajectory arc of a Kühnau quadratic differential Q(z)dz2 that starts at a point
ξ0 ∈ R with degH,Q(ξ0) = N ∈ {0, 1, . . .}.
Theorem 1.2. Suppose that Q(z)dz2 is a Kühnau quadratic differential on H such
that there is a point ξ0 ∈ R with degH,Q(ξ0) = N ∈ {0, 1, . . .}; then we have
Q(w) = (w − ξ0)N
(w − aj)αj
where aj ∈ C and αj ∈ R. Let γ : [0, T ) 7→ ∂H be a simple curve such that
γ(0) = ξ0, γ(0, T ) ⊂ H and γ(0, T ) is a φ-trajectory arc of Q(z)dz2 (φ ∈ [0, π))
that is parameterized by half-plane capacity. Suppose that the functions ft maps
H conformally onto H \ γ(0, t] and are hydrodynamically normalized. Then for
t ∈ (0, T )
(1.1) 2ξ(t) = −µ−C−(t)− µ+C+(t)−
αjAj(t)
+Σ0,
(1.2) ξ̇(t) = − µ
C−(t)− ξ(t) −
C+(t)− ξ(t) −
Aj(t)− ξ(t)
with initial condition ξ(0) = ξ0. Where the functions Aj(t) are defined by
Aj(t) = f
t (aj) for j = 1, . . . , n,
and C+(t) > C−(t) are the two preimages of ξ0 under ft;
µ± = deg
H\γ(0,t],Q(ft(C
±(t))),
Nξ0 +
We can then use Theorem 1.2 to find the driving function in the case when the
slit γ consists of consecutive θk-trajectory arcs of given quadratic differentials. We
will explain how to do this in further detail later. One difficulty with using Theorem
1.2 is that the parameterization is inherently given in terms of half-plane capacity.
This makes it difficult to calculate the driving function ξ if we do not know anything
about the half-plane capacity of the trajectory arc (which, in general, is the case).
The next theorem will allow us to compare the parametrization with the length of
the slit:
6 J. TSAI
Theorem 1.3. Suppose that Q(z)dz2, γ and ft are as defined in Theorem 1.2. Let
Φt(z) =
Q(ft(z))f
(z − ξ(t))2 .
Then γ satisfies
(1.3) γ̇(t) = −2
Φt(ξ(t))
Q(γ(t))
The rest of this paper is organized as follows: In the Section 2, we will state
some basic results from the theory of quadratic differentials and use them to prove
Theorem 1.1. Then we will use Theorem 1.1 to prove Theorems 1.2 and 1.3 in Sec-
tion 3. In Section 4 we will discuss how to obtain the driving function numerically
using Theorems 1.2 and 1.3. Finally in Section 5, we will discuss extensions of The-
orem 1.2 to the case with multiple slits as well as to the radial Loewner differential
equation.
2. Kühnau quadratic differentials and generalized
Schwarz-Christoffel mapping
The aim of this section is to prove Theorem 1.1. We will first look at some of
the basic results in the theory quadratic differentials that we will need.
Transformation Law.
Suppose that f is a conformal map from a domain D2 onto a domain D1 and
suppose that Q1(w)dw
2 is a quadratic differential on D1. If we define
(2.1) Q2(z) ≡ Q1(f(z))f ′(z)2
then Q2(z)dz
2 is a quadratic differential on D2. Then, it is clear that θ-trajectory
arcs are preserved by this transformation law i.e.
γ is a θ-trajectory arc of Q2(z)dz
2 ⇔ f ◦ γ is a θ-trajectory arc of Q2(w)dw2 ,
and also, for z ∈ D2
degQ2(z) = degQ1(f(z)).
Hence trajectories and degQ are conformally invariant in the above sense.
The following lemma tells us that the behaviour of a quadratic differential at a
neighbourhood of a point is determined by the degree of that point.
Lemma 2.1 (Local behaviour of quadratic differentials). Let Q(z)dz2 be a qua-
dratic differential on a domain D. Then for every ω ∈ D there is a conformal
mapping w = φ(z) of some neighbourhood of ω such that
Q(z)dz2 =


dw2 if degQ(ω) = 0,
wndw2 if degQ(ω) = n ≥ 1,
w−ndw2 if degQ(ω) = −n ≤ −1 with n odd,
c2w−2dw2 if degQ(ω) = −2,
(w−n + cw−1)2dw2 if degQ(ω) = −n ≤ −4 with n even.
Here, c is the residue of a branch of
Q(z) at ω.
Proof. See Theorem 8.1 of [13] or Section 6 of [16]. �
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 7
So since trajectories are conformally invariant this lemma tells us that the local
structure of trajectories around a point ω ∈ D is completely determined by degQ(ω)
and the converse is true as well.
Lemma 2.2. Suppose that ω ∈ D and degQ(ω) = n. Then
(1) For n ≥ −1, there are exactly n + 2 trajectories of Q(z)dz2 that end at ω
and form equal angles with each other.
(2) For n ≤ −3, there are infinitely many trajectories ending at ω and more-
over, there are |n| − 2 directions at ω forming equal angles such that the
trajectories approach ω in these directions.
(3) For n = −2, the behaviour depends on the value of c (as defined in Lemma
2.1).
(a) If c is real, then the trajectories are the images of all radial lines under
the map φ defined in Lemma 2.1.
(b) If c is purely imaginary, then the trajectories are the images of all
concentric circles under the map φ defined in Lemma 2.1.
(c) If Re[c], Im[c] 6= 0 then the trajectories are the images of logarithmic
spirals under the map φ defined in Lemma 2.1.
Proof. See Section 7 of [16]. �
This lemma shows that it makes sense for us to define degD,Q(x), the degree of
a point on the boundary, in terms of the trajectories ending at x. If D = H and Q
extends to a meromorphic function on a neighbourhood of some x ∈ R∪ {∞} with
H,Q(x) finite. Then by studying the trajectory structure at x, we can see that
H,Q(x) = degQ(x).
This is the motivation for defining degD,Q in the way we have. The next lemma
shows that the degD,Q is also conformally invariant:
Lemma 2.3. Suppose that Q(z)dz2 is a Kühnau quadratic differential on a domain
D and f is a conformal map of the upper half-plane H onto D. Then the quadratic
differential Qf (w)dw
2 on H, defined by Qf(w) = Q(f(w))f
′(w)2, is also a Kühnau
quadratic differential. Moreover, suppose that z ∈ ∂D is a prime end of D. Then
degD,Q(z) = degH,Qf (f
−1(z)).
Proof. By Carathéodory’s theorem, f extends continuously to ∂H and by Schwarz’s
reflection, f extends analytically across f−1(Γk) for all k = 1, . . . , n. Since θ-
trajectory arcs are conformally invariant, this implies that Qf defined by (2.1) is
a Kühnau quadratic differential on H. Moreover, for all k = 1, . . . , n, f(Γk) is a
θk-trajectory arc of Qf (z)dz
2. Also each point on ∂H corresponds bijectively to a
prime end of H. Hence there is a bijective correspondence between the points of
∂H and prime ends of D. Then
degD,Q(z) = degH,Qf (f
−1(z))
follows from the conformal invariance of trajectories. �
Reflection across trajectories.
Suppose that D is a domain such that Γ ⊂ ∂D is an open interval in R. Let
Q(z)dz2 be a quadratic differential such that Γ is a trajectory arc or an orthogonal
8 J. TSAI
trajectory arc of Q(z)dz2. Then let D− = {z : z ∈ D} be the reflection of D along
Γ. Define
Q−(z) = Q(z) for z ∈ D−.
Then since Γ is a trajectory, we have
Q(z) = Q−(z) ∈ R for z ∈ Γ.
Thus by defining
(2.2) Q∗(z) =
Q(z) for z ∈ D,
Q−(z) for z ∈ D−,
Q(z) = Q−(z) for z ∈ Γ.
it is easy to see that Q∗ is meromorphic in D ∪ D− and hence Q∗(z)dz2 is a
quadratic differential on D ∪ D−. Thus by the transformation law (and using
Schwarz reflection), this shows that we can extend quadratic differentials across
trajectory arcs or orthogonal trajectory arcs.
We will use reflection to prove the following lemma:
Lemma 2.4. Suppose Q(z)dz2 is a Kühnau quadratic differential on H. Then for
any z ∈ ∂H, deg
H,Q(z) ∈ Z implies that Q(z)dz2 extends to a quadratic differential
on a neighbourhood of z and hence
H,Q(z) = degQ(z).
Proof. Firstly, if z ∈ Γj for some j = 1, . . . , n. Then by definition degH,Q(z) = 0
and Q(z)dz2 can be extended to a neighbourhood of z by reflection. By definition,
every z ∈ Γj is an ordinary point of Q(z)dz2 and hence degQ(z) = degH,Q(z) = 0.
Otherwise we write z = zk and suppose that a θk−1-trajectory arc, Γk−1, and a
θk-trajectory arc, Γk, end at zk. Then, by definition, degH,Q(zk) ∈ Z implies that
θk − θk−1 is a multiple of π/2. Thus Γk−1 and Γk are trajectory arcs or orthogonal
trajectory arcs of e−2iθk−1Q(z)dz2. Thus by reflection, e−2iθk−1Q(z)dz2 extends to
a neighbourhood of zk. Hence, Q(z)dz
2 also extends to a neighbourhood of zk. �
We can now prove Theorem 1.1; but first, we explain briefly why we can view
Theorem 1.1 as a generalized form of Schwarz-Christoffelmapping: Schwarz-Christoffel
mapping is a method of computing the conformal map between the upper half-plane
and a domain bounded by a polygon. See [12] for more details. If we have a confor-
mal map f fromH to some domainD such that the sides ofD consist of θ-trajectory
arcs of the quadratic differential Q(w)dw2. Then Q(w)dw2 is a Kühnau quadratic
differential on D and hence by Lemma 2.3, Q(f(z))f ′(z)dz2 is a Kühnau quadratic
differential on H. Theorem 1.1 then implies that
Q(f(z))f ′(z)2 = R
(z − ζj)λj
This is precisely the Schwarz-Christoffel formula when Q(z) ≡ 1.
Also, we comment that the case when Q(w)dw2 is either negative or positive on
R (i.e. the boundary of R consists only of trajectory arcs and orthogonal trajectory
arcs) is easy to prove: we can use reflection to extend Q(z)dz2 to a quadratic
differential on the Riemann sphere Ĉ. Hence Q(z) must be rational since property
(ii) in the definition of Kühnau quadratic differentials guarantees that Q(z) does not
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 9
have any essential singularities and so Q(z) is rational (since the only meromorphic
functions on Ĉ are rational). This proves Theorem 1.1 for this case.
Proof of Theorem 1.1. Since Q(z)dz2 is a Kühnau quadratic differential, we can
z1 < . . . < zm,
(zk−1, zk) for k = 1, . . . ,m,
(zm,∞) for k = m+ 1,
(−∞, z0) for k = 0,
such that each Γk is a θk-trajectory arc of Q(z)dz
2 for some θk ∈ [0, π). Let
T = {Γ1, . . . ,Γm+1} .
Then take any Γ ∈ T . Since Γ is a θ-trajectory for some θ, Γ is a trajectory
arc of e−2iθQ(z)dz2; hence by reflection, we can reflect the quadratic differential
e−2iθQ(z)dz2 across Γ to get a quadratic differential on H− = {Im(z) < 0} which
we call Q̃(z)dz2. Similarly, by rotating Q̃(z)dz2, we can reflect it across another
Υ ∈ T to get another quadratic differential Q∗(z)dz2 on H. Since Q∗ is obtained
from Q by rotating twice, we have
Q∗(z) = eiσQ(z)
for some σ ∈ [0, 2π). This shows that
Ψ(z) =
Q′(z)
(Q∗)′(z)
Q∗(z)
can be extended to a meromorphic function in C\{z1, . . . , zm}. Then part (ii) of the
definition of Kühnau quadratic differentials implies that all the finite singularities of
Ψ(z) are simple poles otherwise Q(z) would have an essential singularity which, by
the great Picard theorem, contradicts part (ii) of the definition of Kühnau quadratic
differentials. Thus we can write:
Ψ(z) = h(z)−
z − ζj
where ζj ∈ C, and λj ∈ R, and h(z) is an entire function in C that does not does
not vanish in C. This implies that
Q(z) = exp
h(ζ)dζ
(z − ζj)λj
(z − zk)νk
Moreover, the singularity at ∞ of
h(ζ)dζ
cannot be essential by part (ii) of the definition of Kühnau quadratic differentials
(otherwise we would get a contradiction with the great Picard theorem as above).
This implies that
h(ζ)dζ
10 J. TSAI
is constant (since it has no zeroes or poles). Hence
Q(z) = R
(z − ζj)λj
If ζj ∈ C \ R, then by definition, we must have
λj = degQ(ζj).
Moreover, if ζj ∈ R and degH,Q(ζj) <∞ we also have
νk = degH,Q(zk).
We will not prove this fact here but in the following corollary we will consider a
special case. The general proof follows readily from it. We will prove the following
corollary which is simply an application of Theorem 1.1 to domains slit by φ-
trajectory arcs:
Corollary 2.5. Suppose that Q(z)dz2 is a Kühnau quadratic differential on H such
that there is a point ξ0 ∈ R with degH,Q(ξ0) = N ∈ {0, 1, . . .}; then we can write
(2.3) Q(w) = R(w − ξ0)N
(w − aj)αj
where aj ∈ C, αj ∈ R, and also R is some non-zero constant. Let γ : [0, T ] 7→ H be a
simple curve such that γ(0) = ξ0 and γ(0, T ) is a φ-trajectory arc of Q(w)dw
2 in H
(φ ∈ [0, π)) and ζ = γ(T ) ∈ H is an ordinary point of Q(w)dw2 (i.e. degQ(ζ) = 0).
Suppose that f maps H conformally onto H \ γ(0, T ]. Then f satisfies
(2.4) Q(f(z))f ′(z)2 = R′(z − ξ)2(z − c−)µ
(z − c+)µ
(z −Aj)αj ,
where R′ is some constant; c−, c+ are the two preimages of ξ0 under f satisfying
c− < c+; Aj is the preimage of aj under f ; and ξ is the preimage of ζ; and
µ± = deg
H\γ(0,s],Q(f(c
Proof. Theorem 1.1 and Lemma 2.4 imply that Q(w) can be written as (2.3). Then
Lemma 2.3 implies that Q̂(z) = Q(f(z))f ′(z)2 is a Kühnau quadratic differential.
So by Theorem 1.1, we only need to look at the singularities of Q̂(z)dz2.
Now, by Schwarz reflection, f extends to a conformal map on C \ {ξ, c−, c+}.
Thus, by the conformal invariance of trajectories, this implies that
Qf (z) = R
′(z − ξ)M (z − c−)µ
(z − c+)µ
(z −Aj)αj .
Then by Lemma 2.2, there are exactly two φ-trajectory arcs of Q(z)dz2 ending
at ζ = γ(T ) of which γ(0, T ) is one of them. So by the conformal invariance of
trajectories, there is one φ-trajectory arcs of Q̂(z)dz2 ending at ξ that is contained
in H. Hence, by definition, deg bQ,H(ξ) = 2. Using Lemma 2.4, this implies that
degQ(ξ) = 2 i.e. M = 2. Thus we only need to determine µ
− and µ+.
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 11
Note that since ξ0 has degree N with respect to Q(z)dz
2, we can determine,
using Lemma 2.2, that the angle between γ(0, T ) and f((ξ, c−)) at ξ0 is
πψ− = π
degH,Q(f(c
−)) + 2
N + 2
and similarly, the angle between γ(0, T ) and f((ξ, c−)) at ξ0 is
πψ+ = π
degH,Q(f(c
+)) + 2
N + 2
Hence, by Schwarz reflection, the function
F (z) = (f(z)− ξ0)1/ψ
extends to a conformal mapping on a neighbourhood of c−. Thus in a neighbour-
hood of z = c−, we can write
(2.5) f(z) = ξ0 + (z − c−)ψ
h(z)ψ
where h is analytic in a neighbourhood of z = c− with h(c−) 6= 0. Now
Q′f (z)
Qf (z)
Q′(f(z))f ′(z)2
Q(f(z))
f ′′(z)
f ′(z)
The residue at z = c− of the left-hand side of the equation is µ−, and we can use
(2.5) to determine the residue at z = c− of the right-hand side. Thus we get
µ− = degH,Q(f(c
We apply the same method to c+ to get µ+. �
3. Domains slit by θ-trajectory arcs
Let Q(w)dw2 be a Kühnau quadratic differential on H with deg
H,Q(ξ0) = N ∈
{0, 1, . . .} for some ξ0 ∈ R. Then by Theorem 1.1 and Lemma 2.4,
Q(w) = (w − ξ0)N
(w − aj)αj
where αj ∈ R and aj ∈ C. Now suppose that γ : [0, T ) 7→ H is a simple curve
such that γ(0) = ξ0 and γ(0, T ) is a φ-trajectory of Q(z)dz
2 in H (φ = [0, π))
that is parameterized by half-plane capacity. As mentioned in the introduction,
there exists conformal maps ft : H 7→ Ht = H \ γ(0, t] satisfying the hydrodynamic
normalization. Then by restricting Q(w)dw2 to a quadratic differential on Ht we
can induce via ft and (2.1), a quadratic differential on H:
(3.1) Qt(z)dz
2 = Q(ft(z))f
2dz2.
We now use Corollary 2.5 and (3.1) to prove Theorem 1.2.
Proof of Theorem 1.2. Note that by Schwarz reflection, each ft can be extended
to a conformal map on Ĉ \ {C−(t), C+(t), ξ(t)}. Then since ft(z) satisfies the
hydrodynamic normalization, this implies that
f ′t(z)
2 = 1 +O
as z → ∞.
12 J. TSAI
So by (3.1),
(3.2)
Qt(z)
Q(ft(z))
= 1 +O
as z → ∞.
If we let ζ = 1/z, then we get
(3.3)
Qt(1/ζ)
Q(ft(1/ζ))
= 1 +O
as ζ → 0.
Since ft is analytic in a neighbourhood of infinity, (3.3) is a Taylor series expansion
and hence we can look at the Taylor series coefficients, in particular:
C(0,ǫ)
f ′(1/ζ)2
C(0,ǫ)
Qt(1/ζ)
ζ2Q(ft(1/ζ))
dζ = 0
for small enough ǫ > 0 where C(0, ǫ) is the anticlockwise contour about the circle
with centre at zero and radius ǫ > 0. Then by Theorem 1.1 and Corollary 2.5, we
can write
Q(w) = R(w − ξ0)N
(w − aj)αj
Qt(z) = R
′(z − ξ)2(z − C−(t))µ
(z − C+(t))µ
(z −Aj(t))αj .
Hence by the residue theorem (since ft(1/ζ) = 1/ζ + · · · as ζ → 0), this implies
2ξ(t) + µ
(t) + µ
(t) +
αkAk(t)
Nξ0 +
This implies (1.1). To get (1.2), note that ft satisfies the chordal Loewner differen-
tial equation (0.1) and hence if we let Gt = f
t ◦ fs for some s ∈ (0, T ) fixed and
t > s, then the chain rule implies that Gt satisfies the differential equation
(z) =
Gt(z)− ξ(t)
Then for some s sufficiently close to t, we can write each Aj(t) = Gt(wj) wj ∈ C
for all j = 1, . . . , n. Thus
Ȧj(t) =
Aj(t)− ξ(t)
Similarly, we get
Ċ±(t) =
C±(t)− ξ(t) .
Hence we get (1.2) from differentiating (1.1). �
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 13
An extension:
We can extend Theorem 1.2 to the case when γ is made up of different θk-trajectory
arcs of some quadratic differential Q(z)dz2: Let γ : (0, T ] 7→ H be a curve with
γ(0) ∈ R such that there is a partition
{0 = t0 < t1 < · · · < tr = T }
such that γ(tk−1, tk) is a θk-trajectory arc of Q(z)dz
2 and γ(tk) is an ordinary
point of Q(z)dz2 for k = 1, . . . , r. Then we can find the driving function ξ(t) of γ
by applying Theorem 1.2 to the θ1-trajectory arc γ(0, t1) to get a driving function
ξ1(t), and applying Theorem 1.2 inductively to each f
(γ(tk, tk+1)) (which is a
θk+1-trajectory arc of the quadratic differential Qtk(z)dz
2 = Q(ftk(z))f
(z)2dz2)
to get ξk(t). Then
ξ(t) = ξk(t) for t ∈ [tk−1, tk).
We also have the following corollary:
Corollary 3.1. Suppose that Q(w)dw2 and γ are as defined in Theorem 1.2.
Then the driving function ξ and Aj , C
−, C+ as defined in Theorem 1.2 are in
C∞(0, T ). Moreover, we can write any derivative of ξ, C−, C+, Aj explicitly in
terms of ξ, C−, C+, Aj and the exponents µ
−, µ+, αj.
Proof. Recall that, in the proof of Theorem 1.2, we had the formulae
Ȧj(t) =
Aj(t)− ξ(t)
, Ċ±(t) =
C±(t)− ξ(t) .
This implies that each term in (1.2) is differentiable so we can write the second
derivative of ξ in terms of ξ(t), Aj(t), C
±(t). This in turn implies that we can write
the third derivative of ξ in terms of ξ(t), Aj(t), C
±(t) and the exponents. Continuing
inductively, we have showed that every derivative of ξ exists and can be expressed
in terms of ξ(t), Aj(t), C
±(t) and the exponents. Note that each derivative of ξ is
finite for t ∈ (0, T ) since
|Aj(t)− ξ(t)|, |C±(t)− ξ(t)| > 0.
Then ξ is smooth implies that Aj(t), C
±(t) are also smooth. �
Theorem 1.3 then follows from Corollary 2.5 and Theorem 1.2:
Proof of Theorem 1.3.
First note that, by the definition of θ-trajectory arcs, γ̇ always exists and is never
0. Also by Corollary 2.5, Φt(ξ(t)) 6= 0,∞; thus the right hand side of (1.3) always
exists since, by definition, γ avoids poles and zeroes of Q(w)dw2.
Recall that ft(ξ(t)) = γ(t), this implies that
γ̇(t) = ḟt(ξ(t)) + f
t(ξ(t))ξ̇(t).
Then combining the Loewner differential equation (0.1) with (3.1) we have
ḟt(z) = −
z − ξ(t)
Qt(z)
Q(ft(z))
Φt(z)
Q(ft(z))
⇒ ḟt(ξ(t)) = −2
Φt(ξ(t))
Q(γ(t))
Note that Φt(ξ(t)) 6= 0,∞ since, by Corollary 2.5, Qt(z) has a double zero at ξ(t).
14 J. TSAI
Thus we have
γ̇(t) = −2
Φt(ξ(t))
Q(γ(t))
Qt(ξ(t))
Q(γ(t))
ξ̇(t) = −2
Φt(ξ(t))
Q(γ(t))
since Theorem 1.2 implies that ξ̇ is finite for all t ∈ (0, T ) and Corollary 2.5 implies
that Qt(ξ(t)) = 0. �
4. Applying Theorem 1.2
In practice, understanding ξ(t) via (1.1) is not possible: it is difficult to calculate
the positions of the zeroes and poles of Qt because the information we have on them
is all relative to ξ(t) (which we are trying to find). On the other hand, (1.2) is more
useful in applications. In this section, we will demonstrate how we can use (1.2)
to calculate numerically the driving function of a given slit that consists of θk-
trajectory arcs of a given quadratic differential. The method is basically a modified
version of Euler’s method.
Firstly, for any smooth function h on (0,T), Taylor’s theorem implies that for all
M = 1, 2, . . .,
(4.1)∣∣∣∣∣h
h(t) +
)∣∣∣∣∣ ≤
M !KM
s∈(t,t+ 1K )
for t, t + 1/K ∈ (0, T ). We will apply (4.1) to the functions ξ, Ak and C± (as
defined in Theorem 1.2) noting that, by Corollary 3.1, they are smooth and all of
their derivatives can be expressed in terms of ξ(t), Ak(t) and C
±(t). Thus if we
know ξ(s), Ak(s) and C
±(s) we can use (4.1) to obtain an approximate formula for
ξ(s+K−1), Ak(s+K
−1) and C±(s+K−1) (choosing K to be small and/or M to
be large so that the right-hand-side of (4.1) is small); then we can apply (4.1) to
ξ(s+K−1), Ak(s+K
−1) and C±(s+K−1) to find ξ(s+2K−1), Ak(s+2K
−1) and
C±(s+2K−1). Continuing like this, we obtain an approximation of ξ at the points
{s+ nK−1}.
So clearly what we need to do now is find the starting values ξ(s), Ak(s) and
C±(s) so we can apply the above method. But because ξ is not differentiable at
0, we cannot use the formula (4.1) with t = 0. The way around this is to note
that if deg
H,Q(ξ0) = N ∈ {0, 1, 2, . . .} then since we know degH,Qt(C
+(t)), we can
calculate the angle that the trajectory makes with the line [ξ0,∞) (as in the proof
of Corollary 2.5). Then we find that the angle is πψ where:
πψ = π
2 deg
(C+(t)) + 2
N + 2
So if we choose s small enough, we have
fs ≈ Fψ,ξ0s ,
where Fψ,ξ0s is the conformal map that maps H conformally onto H
s that is hydro-
dynamically normalized where Hψs is the upper half-plane slit by the straight line
starting at ξ0 making an angle πψ with [ξ0,∞), with half-plane capacity 2s. Then
we also have
Ak(s) ≈ (Fψ
−1(ak),
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 15
and also, C−(s), C+(s) are approximately the two preimages of ξ0 under F
ψ+,ξ0
Then we can use (1.1) to calculate ξ(s) approximately. We can then plug this
information into (4.1) as described above.
Note that F
t can be found using the fact that
(4.2) F
λt (z) =
z − (1− 2p)
z − (1− 2p)
t+ (1− p)
for some λ. Then we reparameterize this formula to remove the λ and translate the
point 0 to ξ0. Unfortunately, inverting this function cannot be done explicitly but
it can be done numerically very efficiently using Newton’s method. Alternatively,
by selecting a small s, we can assume that
Ak(s) ≈ ak
for all k. Then we note that the 2 preimages of ξ0 under F
s can be determined
explicitly (see [11]). This obviates the need to numerically invert F
Another difficulty is that, in general, given a slit, we cannot parameterize it by
half-plane capacity so it would be difficult, for example, to know at which t one
should stop. Most formulae for calculating half-plane capacity of some compact set
K rely on knowing the conformal map fK of H onto H\K (normalized hydrodynam-
ically). One possibility would be to use the probabilistic definitions of half-plane
capacity given in [7]. We will use the fact that Theorem 1.3 and Corollary 3.1 imply
that we can give all derivatives of γ(t) in terms of ξ(t), Ak(t), C
−(t), C+(t) and the
exponents µ−, µ+, αk so if we know these, we can also use (4.1) to approximate
γ. This in turn allows us to calculate the length of the slit γ. Thus if we know
beforehand length of our slit, we can calculate at what value of t we stop.
We now have everything we need in order to use (4.1) to calculate the driving
function numerically of any slit that is made up of θk trajectory arcs of a qua-
dratic differential Q(w)dw2. We will demonstrate how this is done in the following
example:
An example. Suppose that γ : (0, T ) → H is a piecewise linear arc parameterized
by half-plane capacity that satisfies:
• γ(0) = 0.
• From t = 0 to t = t1, γ is the straight line arc from 0 to i; call this Γ1.
• From t = t1 to t = t2, γ is the straight line arc from i to 2 + i; call this Γ2.
• From t = t2 to t = t3 = T , γ is the straight line arc from 2 + i to 2 + 2i;
call this Γ3.
First note that γ is made up of alternating (π/2)- and 0-trajectory arcs of the qua-
dratic differential 1dw2 in H and hence we can use Theorem 1.2 (or more specifically
the extension of Theorem 1.2 detailed in Section 3) to calculate ξ̇. As mentioned
previously, there is no easy way to know beforehand what t1, . . . , t3 are. For sim-
plicity, we will only use M = 1 in (4.1) i.e.
≈ f(t) + ḟ(t)
and fix a large K. Obviously Γ1 forms a right angle with real line; so we can use
(4.2) to determine the function
ft1 = F
1/2,0
(z) =
z2 − 4t1.
16 J. TSAI
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
0 1 2 3 4 5 6
Figure 3. The example path in the upper half-plane (left) and a
plot of its driving function on the y-axis against time on the x-axis
(right).
It is easy to see that in this case, t1 = 1/4 and ξ is constantly 0 for t ∈ (0, t1].
This induces the quadratic differential using (3.1):
Qt1(z)dz
z2dz2
(z + 1)(z − 1) .
Hence, we let A1(t1) = −1, A2(t1) = 1. Also f−1t1 (γ2) is a 0-trajectory arc of
Q1(z)dz
2 starting from ξ(t1) = 0 on R (by the conformal invariance of trajectories).
Now note that f−1t1 (γ2) makes an angle of π/4 with the positive real axis. and hence
ft1+K−1 ≈ F
1/4,0
sinceK is large. We can then use Newton’s method to find the preimages under the
above approximation of ft1+K−1 of the points A1(t1), A2(t1) and the 2 preimages of
zero to get the points A1(t1+K
−1), A2(t1+K
−1), C−(t1+K
−1), C+(t1+K
−1) and
hence, using (1.1), we can find ξ(t1+K
−1). Then inserting this into (4.1), as detailed
above we can also find ξ(t1 + nK
−1) and A1(t1 + nK
−1), A2(t1 + nK
−1), C−(t1 +
nK−1), C+(t1 + nK
−1); also, by Theorem 1.3, we can find |γ̇(t1 + nK−1)| if we let
t2(K) = inf
|γ̇(t1 + nK−1)| > (length of Γ2) = 2
then t2(K) ≈ t2 for K large. So we just assume that t2 = t2(K). Let A3(t2) =
C−(t2) and A4(t2) = C
+(t2). Hence by (3.1),
Qt2(z)dz
(z − ξ(t2))2(z −A3(t2))dz2
(z −A1(t2))(z −A2(t2))(z −A4(t2))
Then, by the conformal invariance of trajectories, f−1t2 (Γ3) is a π/2-trajectory of
Qt2(z)dz
2 and also, f−1t2 (Γ3), makes an angle 3π/4 with (ξ(t2),∞) and so
ft2+K−1 ≈ F
3/4,0
Then, as before, we can use Newton’s method to find the preimages under the above
approximation of ft2+K−1 of the points A1(t2), . . . , A4(t2) and the 2 preimages of
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 17
ξ(t2) to get the points A1(t2+K
−1), . . . , A4(t2+K
−1), C−(t2+K
−1), C+(t2+K
and hence use (1.1) to get ξ(t2+K
−1). We insert these into the formula iteratively
to get ξ(t2+nK
−1) and A1(t2+nK
−1), . . . , A4(t2+nK
−1), C−(t2+nK
−1), C+(t2+
nK−1) until t2+nK
−1 ≈ T . Thus the end result is that we found the driving func-
tion of the first 3 steps of the slit given in Figure 3. Of course, our calculation of ξ
will be more accurate by taking larger K.
For example, we can use the above method to calculate the driving function of
any path on the square/triangle/hexagonal lattice on H starting from some point in
R. In fact we can calculate the driving function of a path on the square/triangle/hexagonal
lattice in any polygon D by mapping the half-plane conformally onto D and pulling
back the quadratic differential 1dw2 on D to Q(z)dz2 on H using the transforma-
tion law. Also note that, in general, any curve γ can be approximated by a curve
γδ which lies on the square lattice δZ
2. Then it can be shown that
ξδ → ξ uniformly as δ ց 0,
where ξδ is the driving function of γδ and ξ is the driving function of γ hence, we
can use the above method to calculate ξδ then take the limit as δ ց 0 to obtain ξ.
Another point to note is that using the above method, we do not need to know
before hand what the trajectory arc of the given quadratic differential looks like;
so for arbitrary Kühnau quadratic differentials, we can use this method to plot the
trajectories starting at the boundary.
We end this section by looking at what happens when the slit approaches the
boundary:
Proposition 4.1. Suppose that γ : [0, T ) 7→ H is a simple curve such that γ(0) ∈ R
and γ(0, T ) is a θ-trajectory arc of some quadratic differential Q(z)dz2. Then let ξ
be the driving function of γ. If
γ(t) ∈ R ∪ γ(0, T ),
i.e. γ makes a loop at time T . Then
∣∣∣∣→ ∞ as tր T
for all n = 0, 1, . . ..
Proof.
For t ∈ (0, T ), we define
Γ(t) = {γ(s) : s ∈ (t, T )}.
Then Γt is a θ-trajectory arc in Ht = H\γ(0, t] of Q(w)dw2 and it is also a crosscut
in Ht (see [14]). Then by the conformal invariance of θ-trajectories, f
t (Γt) ⊂ H
is a θ-trajectory arc of Qt(z)dz
2. Moreover, f−1t (Γt) is a crosscut of H with one
end point at ξ(t) and the other end point in R such that either C+(t) or C−(t)
is contained in the closure of the bounded component of H \ f−1t (Γt). Without
loss of generality, assume it is C+(t). Then since diam(f−1t (Γt)) → 0 as t ր T ,
we must have ξ(t) = C+(T ) and hence by (1.2), ξ̇(t) → ∞ as t ր T . Similarly,
we differentiate (1.2) as mentioned in Corollary 3.1 to obtain the result for higher
order derivatives. �
18 J. TSAI
This means that as γ gets closer and closer to making a loop, the approximation
by (4.1) stops working no matter what M we choose. This phenomenon can be
observed in Figure 3, as we turn the last corner in γ, we can see that ξ decreases
faster even though the slit is not yet that close to the boundary.
5. Generalizing Theorem 1.2
5.1. Multiple slits. Suppose that γk : [0, T ) → H for k = 1, . . . , N are disjoint
simple curves such that γk(0) ∈ R and γk(0, T ) ⊂ H. By the Riemann mapping
theorem, there exists unique ft that map H conformally onto Ht = H\
k=1 γk(0, t]
that satisfies the hydrodynamic normalization. We can reparameterize such that
γk(0, T ]
has half-plane capacity 2t. Then ft satisfies
(5.1) ḟt(z) = −2f ′t(z)
bk(t)
z − ξk(t)
where
bk(t) = 1,
and ξk(t) = f
t (γk(t)). See [2] for more details.
Theorem 5.1. Suppose that Q(w)dw2 is a Kühnau quadratic differential on H
such that the points ξk(0) ∈ R satisfy
H,Q(ξk(0)) = βk ∈ {0, 1, 2, . . .}
for all k. Then we can write
Q(w) =
(w − ξk(0))βk
(w − aj)αj
with aj ∈ C and αj ∈ R. Then suppose that γk : [0, T ) → H for k = 1, . . . , N are
disjoint simple curves such that γk(0) ∈ R and γk(0, T ) ⊂ H and are parameterized
as above. Then
(5.2) 2
ξk = −
(µ−k C
k (t) + µ
k (t))
αjAj(t)
+Σ0,
(5.3)
ξ̇l(t) =
k=1,k 6=l
bk(t)
ξl(t)− ξk(t)
bl(t)
(t)− ξl(t)
bl(t)
(t)− ξl(t)
αjbl(t)
Aj(t)− ξl(t)
for all l ∈ {1, . . . , N}. Where C−k (t) and C
k (t) are the two preimages of ξk(0)
under ft satisfying C
k (t) < C
k (t);
µ±k = degHt,Q(ft(C
k (t)));
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 19
Aj(t) = f
t (aj); and
βkξk(0)
Proof. By Theorem 1.1 and Lemma 2.4, we can write
Q(w) =
(w − ξk(0))βk
(w − aj)αj
Then either by modifying the proof of Corollary 2.5 or iterating N slit functions
and applying Corollary 2.5 N times, it is not too difficult to see that if we define
Qt(z) by (3.1), then
(5.4)
Qt(z) =
(z − ξk(t))2(z − C−k (t))
k (z − C+k (t))
(z −Aj(t))αj
Then the proof of (5.2) is exactly the same as the proof of (1.1) in Theorem 1.2.
The proof of (5.3) is more complicated. First let
Pt(z) = −2
bk(t)
z − ξk(t)
Then (5.1) becomes
ḟt(z) = f
t(z)Pt(z).
Now take the logarithmic derivative of Qt(z) with respect to z and t separately
using the definition of Qt(z) given by (3.1) to get
Q′t(z)
Qt(z)
Q′(ft(z))f
Q(ft(z))
f ′′t (z)
f ′t(z)
Q̇t(z)
Qt(z)
Q′(ft(z))ḟt(z)
Q(ft(z))
ḟ ′t(z)
f ′t(z)
Q′(ft(z))f
t(z)Pt(z)
Q(ft(z))
f ′′t (z)Pt(z) + f
t(z)P
t (z)
f ′t(z)
where we substitute (5.1) in for ḟt to get from the first to the second line. Thus we
(5.5)
Q̇t(z)
Qt(z)
Q′t(z)
Qt(z)
Pt(z) + 2P
t(z).
So then by (5.4), we note that
Q′t(z)
Qt(z)
z − ξk(t)
z − C−k (t)
z − C+k (t)
z −Aj(t)
Q̇t(z)
Qt(z)
2ξ̇k(t)
z − ξk(t)
µ−k Ċ
k (t)
z − C−k (t)
µ+k Ċ
k (t)
z − C+k (t)
αjȦj(t)
z −Aj(t)
20 J. TSAI
0 2 4 6 8 10
Figure 4. A plot of the two driving functions ξ1 (top) and ξ2
(bottom) on the y-axis against time on the x-axis.
Thus substituting this into (5.5) and comparing the coefficient of
z − ξl(t)
(i.e. the residue at z = ξl(t) of both sides of (5.5)), we find that this is exactly
(5.3). �
Similarly, we can prove a version of Theorem 1.3 and Corollary 3.1 for multiple
slits. This means that we can use the method detailed in Section 4 with (5.3)
to calculate the driving function for multiple θk-trajectory arc slits. For example
Figure 4 plots the graph of the two driving functions ξ1 and ξ2 in the case when
γ1 and γ2 are 2 vertical slits starting from -1 and 1 (i.e. orthogonal trajectories of
1dz2) and growing at the same speed. Compare this with Figure 7 in [2].
5.2. Radial Loewner evolution. The chordal Loewner differential equation was
introduced because the upper half-plane was an easier domain to work with for
many applications but the original setting of the Loewner differential equation is in
the unit disc D = {z ∈ C : |z| < 1}: Suppose that γ : [0, T ) 7→ D is a simple curve
such that γ(0) ∈ T = {z : |z| = 1} and γ(0, T ) ⊂ D \ {0}. Then Dt = D \ γ(0, t]
is simply-connected and 0 ∈ Dt for all t ∈ (0, T ). Hence the Riemann mapping
theorem implies that there is unique conformal map ft mapping D conformally onto
Dt such that ft(0) = 0 and f
t(0) > 0. Then Schwarz’s lemma and the Carathéodory
kernel theorem implies that f ′t(0). is strictly decreasing and continuous so we can
reparameterize such that f ′t(0) = e
−t. f ′t(0) is sometimes called the conformal
radius of Dt; hence in this case we are parameterizing by conformal radius. Then
the functions ft satisfy the radial Loewner differential equation:
ḟt(z) = −zf ′t(z)
z + eiξ(t)
z − eiξ(t)
See [9] for more details.
THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 21
Theorem 5.2. Suppose that Q(w)dw2 is a Kühnau quadratic differential on D such
that degQ(0) = K ∈ Z and eiξ0 ∈ T = {|z| = 1} satisfies
= N ∈ {0, 1, 2, . . .}
then we have
Q(w) = wK(w − eiξ0)N
(w − aj)αj
where aj ∈ C and αj ∈ R. Then if γ : [0, T ) 7→ D is a simple curve such that
γ(0) = eiξ0 and γ(0, T ) ⊂ D \ {0} is a φ-trajectory arc of Q(w)dw2 in D that does
not meet 0 and is parameterized as above. Then we have
(5.6) e2iξ(t) = e−2tΠ0C
−(t)−µ
C+(t)−µ
Aj(t)
−αj ,
(5.7)
ξ̇(t) = − 1
C−(t) + eiξ(t)
C−(t)− eiξ(t)
C+(t) + eiξ(t)
C+(t)− eiξ(t)
Aj(t) + e
iξ(t)
Aj(t)− eiξ(t)
where, as usual, the functions Aj(t) are defined by
Aj(t) = f
t (aj) for j = 1, . . . , n,
µ± = degDt,Q(ft(C
±(t))),
C+(t) > C−(t) are the two preimages of eiξ0 under ft; and also,
Π0 = e
Proof. The formula for Q(w) can be obtained from Theorem 1.1 by the transforma-
tion law. We then define Qt by (3.1). Since the point 0 is fixed by ft, this implies
that the degQt(0) = degQ(0) = K. Thus we can apply Corollary 2.5 (again, using
the transformation law) to get
Qt(z) = z
K(z − eiξ(t))2(z − C−(t))µ
(z − C+(t))µ
(z −Aj(t))αj
Then since by definition,
Qt(z) = Q(ft(z))f
This immediately implies (5.6) by substituting z = 0. Then we get (5.7) in the
same way as we get (1.2) from (1.1) in the proof of Theorem 1.2. �
As in the case of multiple slits, a version of Theorem 1.3 and Corollary 3.1 holds
for this case.
22 J. TSAI
5.3. Other versions of the Loewner differential equation. There are several
other versions of the Loewner differential equation for simply-connected domains in
the literature; the methods in this paper should work in those cases as well and the
proofs should be similar to the proofs of Theorem 1.2 etc. Also, [4], [5] generalizes
the Loewner differential equation to multiply-connected domains and again, some
of the methods should work in these cases possibly using methods in [1] to extend
Theorem 1.1 to multiply-connected domains. Finally, even if we consider general
2-dimensional growth processes given by the Loewner-Kufarev differential equation
(see Chapter 6 of [13]), some of the methods in this paper should still be applicable.
Acknowledgement. The author would like to express his gratitude to his super-
visor Dr. T. K. Carne for his constant guidance and helpful discussion when writing
this paper.
References
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Phys., 115(3-4):805–822, 2004.
3. T. Kennedy. Computing the Loewner driving process of random curves in the half plane.
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12. Z. Nehari. Conformal mapping. Dover Publications, New York, 1982.
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Lehrbücher, Band XXV.
14. C. Pommerenke. Boundary behaviour of conformal maps. Springer-Verlag, Berlin, 1992.
15. O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel
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Department of Mathematics, Chinese University of Hong Kong, Shatin, New Terri-
tories, Hong Kong
E-mail address: jtsai@math.cuhk.edu.hk
http://arxiv.org/abs/math/0702071
	Introduction
	1. Main results
	Definition:
	2. Kühnau quadratic differentials and generalized Schwarz-Christoffel mapping
	Transformation Law
	Reflection across trajectories
	3. Domains slit by -trajectory arcs
	An extension:
	4. Applying Theorem ??
	An example.
	5. Generalizing Theorem ??
	5.1. Multiple slits
	5.2. Radial Loewner evolution
	5.3. Other versions of the Loewner differential equation
	Acknowledgement.
	References
