Constant Coefficients in the Radial Komatu-Loewner Equation for Multiple Slits

Constant Coefficients in the Radial Komatu-Loewner Equation for Multiple Slits

Christoph Böhm and Sebastian Schleißinger
July 12, 2019
Abstract

The radial Komatu-Loewner equation is a differential equation for certain normalized conformal mappings that can be used to describe the growth of slits within multiply connected domains. We show that it is possible to choose constant coefficients in this equation in order to generate given disjoint slits and that those coefficients are uniquely determined under a suitable normalization of the differential equation.

1 Introduction and results

In 1923, C. Loewner derived a differential equation for a certain family of conformal mappings to attack the Bieberbach conjecture, see [8]. Loewner’s method has been extended and turned out to be a useful tool within complex analysis. In particular, the Loewner differential equations provide a powerful tool for the description of the growth of slits in a given planar domain. After O. Schramm discovered Stochastic Loewner Evolution (or Schramm Loewner Evolution, SLE) in [11], it became clear that those models have many applications in different mathematical and physical disciplines, especially in statistical physics.

In the classical setting, Loewner theory describes the evolution of a family of simply connected, proper subsets of the complex plane which are all conformally equivalent to the unit disc according to the Riemann Mapping Theorem. In 1943, Y. Komatu showed that Loewner’s ideas are not confined to the simply connected case: He derived a Loewner equation for the growth of a slit within a doubly connected domain, see [6]. In [7], he considered a generalization of Loewner’s differential equation to a more general finitely connected domain. Komatu’s ideas have been applied and extended by several authors. Recently, R. Bauer and R. Friedrich derived a radial and a chordal Komatu-Loewner equation to deal with the growth of a (stochastic) slit in a multiply connected domain, see [1] and [2]. In the radial setting, the slit grows within a circular slit disk , i.e. , where each is a circular arc centered at such that whenever Note that every -connected domain can be mapped onto such a circular slit disk by a conformal map This mapping is unique if we require the normalization for some see [4], Chapter 15.6.

In [3], W. Lauf and the first author generalized the radial Komatu-Loewner equation for the growth of several slits:
Let be an arbitrary circular slit disk and let be parametrizations of pairwise disjoint simple curves such that and for all
Furthermore, if is a circular slit disk and , we denote by the unique conformal mapping from onto the right half-plane minus slits parallel to the imaginary axis with and
We summarize one of the main results of [3] in the following theorem:

Theorem A (Corollary 5 in [3]).

Let and denote by the unique conformal mapping where is a circular slit disk and
Then there exists a Lebesgue measure zero set such that for every the function is differentiable on with

(1.1)

where the continuous function is the image of under the map and the coefficient functions are measurable with for every .

Remark 1.

The continuous functions are usually called driving functions.
Informally, the coefficient function corresponds to the speed of growth of the slit parametrized by From the normalization it follows that

Figure 1: The mappings and from Theorem A.

Note that there are no further assumptions on the parametrizations of the slits in Theorem A.
Now suppose we are given only the circular slit disk and the slits without parametrization. Roughly speaking, we address the question if it is possible to find a simple form of equation (1.1) such that it has still enough parameters to generate the slits , but, on the other hand, the choice of those parameters is unique. We will see that this is possible and, moreover, it will turn out that equation (1.1) is satisfied for all in this case.

The latter can be interpreted as a generalization of Loewner’s original idea of finding a parametrization of an arbitrary curve such that the family of certain associated conformal mappings is differentiable. In some sense, this problem is related to Hilbert’s fifth problem of finding differentiable structures for continuous groups, see [5].

In a first step, we can choose parametrizations such that the mappings satisfy i.e. However, there are many of such parametrizations when .

In this work we will show that there exist unique parametrizations of the slits , such that the corresponding Komatu-Loewner equation is satisfied for all where all coefficient functions are constant and sum up to 1.

We need one further notation: Let be a circular slit disc and let be the unique conformal mapping onto a circular slit disk with and , then and the logarithmic mapping radius of is defined to be the real number . The inequality is an immediate consequence of Lemma 3 b).

Theorem 1.

Let be the logarithmic mapping radius of . There exist unique continuous parametrizations

and unique with such that the following holds:
Let and denote by the unique conformal mapping where is a circular slit disk and
Then and for every the function is differentiable on with

(1.2)

where is the image of under the map . Moreover, the driving functions are continuous.

In the simply connected case, we have for all and equation (1.2) can be written down explicitly as

Furthermore, coefficients and continuous driving functions determine the unique solution to equation (1.2) in the simply connected case. If the solution generates slit mappings, then the parametrization of these slits are uniquely determined by the driving functions and the coefficients.

Thus we can formulate the simply connected case of Theorem 1 as follows.

Corollary 2.

Let be disjoint slits in and let be the logarithmic mapping radius of . Then there exist unique and unique continuous driving functions such that the solution of the Loewner equation

generates the slits i.e. maps conformally onto .

Remark 2.

In [9], D. Prokhorov has proven the existence and uniqueness of constant coefficients for several slits in the simply connected case under the assumption that all slits are piecewise analytic. This theorem forms the basis for Prokhorov’s study of extremal problems for univalent functions in [9] by using control-theoretic methods. Our proof shows that one can drop any assumption on the regularity of the slits in order to generate them with constant coefficients.

We shall give the details of the proof only in the case , i.e. for two slits, in order to allow a simple notation. The general case of slits can be proved inductively in exactly the same way.
Our proof combines some technical tools from [3] that were used to prove Theorem A and an idea from [10], where a similar result was proven for the so called chordal Loewner equation in the simply connected case.
The rest of this paper is organized as follows. In Section 2 we describe the setting for the proof and cite some technical results from [3]. The proof of Theorem 1 is divided into two parts: In Section 3 we prove the existence statement and in Section 4 we give the proof of the uniqueness statement of Theorem 1.

2 The setting for the proof

Suppose is a circular slit disk and , are disjoint slits, i.e. there are continuous, one-to-one functions with such that , and for
Let the function be defined as in Theorem A. As is monotonically increasing (see Lemma 3) and , we can assume without loss of generality that Otherwise, we can simultaneously reparameterize the two slits. Consequently we have , where denotes the logarithmic mapping radius of .

Furthermore, we assume that in order to simplify some of the notations.

For every we let be the unique conformal mapping from onto a circular slit disk with and and we define

Now, in order to prove Theorem 1, we have to show that there exist

  • two uniquely determined increasing homeomorphisms and

  • a uniquely determined

such that the Komatu-Loewner equation for the slits and is satisfied for all with and and for all

First, we summarize some basic properties of lmr in the following lemma.

Lemma 3.

  1. The function is continuous in .

  2. The function is strictly increasing with respect to and respectively.

  3. For every there exists a so that for all and with and the following holds:

Proof.

See Proposition 6, 8 and 15 in [3]. ∎

Furthermore, we will use a dynamic interpretation of the coefficient functions from Theorem A. Let be arbitrary strictly increasing homeomorphisms and let be a partition of the interval for a fixed , i.e. We will denote by the norm of Now we define the two sums

The following proposition relates the limit of for to the coefficient functions of Theorem A.

Proposition 4.

Let be two increasing self-homeomorphisms and let denote the conformal mapping for the slits and from Theorem A. Assume that , i.e. for all . Then the limits

exist and form two increasing and Lipschitz continuous functions with and . Furthermore, if is differentiable in for and then the differential equation (1.1) holds for with

In this case, is equal to the first derivative of the function in and is equal to the first derivative of the function in .

Proof.

This follows immediately from Proposition 16 in [3], Theorem 2 in [3] and the definition of in [3]. ∎

Beside and we define for a partition of the interval

If holds for every , then it is easy to see that and . By Proposition 4 we see

3 Existence

Now we are able to prove the existence part of Theorem 1. Recall that we have to show the existence of two strictly increasing homeomorphisms and a such that the Komatu-Loewner equation for the slits and is satisfied for all with and and for all

The proceeding of this proof is as follows.

  1. First of all we will use a Bang-Bang method introduced in [10] to construct two sequences and of increasing self-homeomorphisms of .

  2. By using a diagonal argument on and we will find two subsequences and which converge pointwise on a dense set to increasing functions and respectively. The functions and can be extended to continuous functions defined on , with . Furthermore, we will get by the construction of and .

  3. Next we will derive a connection between the sum and the sum for a given partition of the interval .

  4. Moreover, we will find a connection between and .

  5. By combining these results we will find if . Furthermore, as a consequence of this, we will find .

  6. Next will show that and are strictly increasing, i.e. both functions are increasing self-homeomorphisms of

  7. Finally we will obtain the Komatu-Loewner-Equation with constant coefficients and for the parametrizations and .

Proof of Theorem 1 (Existence).

  • To construct and , we first extend both and to an interval such that and are still disjoint slits and Let and . We let and for we define and recursively as the unique values with

    Since is strictly increasing in both variables, see Lemma 3 b), we get

    Consequently .
    Furthermore, note that the values and depend continuously on : This follows easily by induction and the continuity and strict monotonicity of the function , see Lemma 3. Consequently, for every , we can find a value with . Now we define a sequence of functions and . Define

    for all . The values of and between the supporting points are defined by linear interpolation. An immediate consequence of this construction is

    (3.1)
  • Since is bounded, we find a subsequence such that is convergent with the limit . Next we set

    is a dense and countable subset of . Denote by a bijective mapping.
    Since the sequences and are bounded (by ), we find a subsequence of , so that and are convergent.

    Inductively, we define , , to be a subsequence of such that and are convergent.

    Consequently we can define sequences and which are (pointwise) convergent in . We denote by and the limit function, i.e.

    for all . Moreover we set and . By using equation (3.1) we get for if is big enough. Consequently we find by using Lemma 3 a)

    (3.2)

    for all . Furthermore, since and are strictly increasing, the functions and are increasing too. Moreover and can be extended in a continuous and unique way to [0,1]. To see this, let and define

    Thus we find by Lemma 3 a) and equation (3.2)

    Since is strictly increasing in both variables and and , we find and . If we can argue in the same way, so and are continuous in . Summarizing, and are continuous and increasing in [0,1] with and . For later use we define and .

  • Next we show that for every fixed , fixed and a fixed partition of the interval , there exists an so that

    holds for all , where .

    Fix . As the function is (uniformly) continuous in by Lemma 3 a), there exists such that

    Since , we find an so that , holds for all and . Consequently we find

  • For now we fix . We show that for all we find a so that for all partitions of with there exists an so that for all we have

    Let . Then there exists such that the inequality from Lemma 3 c) holds. Since the functions and are (uniformly) continuous we get

    Denote by a partition of with and . Then we find an with , and

    for all and all . As a consequence we get

    for all and all . In an analog way we get . Next, we set . is a partition of the interval and we write .

    where if with and . Since , we have for all and all . Thus we get

    Since and for all and all , we have by Lemma 3 c)

    for all . The last inequality can be proven by using the monotonicity of as follows