The Schramm–Loewner equation for multiple slits
We prove that any disjoint union of finitely many simple curves in the upper half–plane can be generated in a unique way by the chordal multiple–slit Loewner equation with constant weights.
1 Introduction and results
Recent progress in the theory of Loewner equations ([Loe23, Sch00, Law05, BCD12]) suggests that one of the most useful descriptions of a simple plane curve is by encoding it into a growth process modeled by the Schramm–Loewner equation. In this paper we show that any disjoint union of finitely many simple curves can be encoded in a unique way into a growth process described by a multi–slit version of the Schramm–Loewner equation. In order to state our result we need to introduce some notation.
Let be the upper half–plane. A slit is the trace of a simple curve with and . Since is a simply connected domain, (a version of) Riemann’s mapping theorem guarantees that there is a unique conformal map from onto with hydrodynamic normalization
for some . We call the half–plane capacity of the slit . The following well–known result provides a description of the slit with the help of the chordal Loewner equation (Schramm–Loewner equation).
Theorem A (The one–slit chordal Loewner equation)
Let be a slit with . Then there exists a unique continuous driving function such that the solution to the chordal Loewner equation
has the property that .
Note that in Theorem A the slit “starts” at the point .
To the best of our knowledge the first proof of Theorem A is due to Kufarev, Sobolev, Sporyševa in [KSS68]. The basic recent reference for Theorem A is the book of Lawler [Law05]. We also refer to the survey paper [GM13] for a complete and rigorous proof of Theorem A using classical complex analysis.
Now, let be a multi–slit, that is, the union of slits with disjoint closures. As before, there is a unique conformal map from onto with expansion
and we call the half–plane capacity of . The main result of the present paper is the following extension of Theorem A.
Let be a multi–slit with . Then there exist unique weights with and unique continuous driving functions such that the solution of the chordal Loewner equation
Some remarks are in order.
Remark 1.2 (The multi–slit chordal Loewner equation)
continuous weight functions with and for every , and
continuous driving functions ,
such that the solution to the multi–slit chordal Loewner equation
satisfies , see Remark 2.1 below. However, the functions and are not uniquely determined by the multi–slit if , simply because in this case there are obviously many Loewner chains (in the sense of [Law05]) such that . Informally, each weight function corresponds to the speed of growth of the slit (the one that starts at the point ). Theorem 1.1 shows that one can actually choose constant weight functions , which are moreover uniquely determined. In addition, then also the driving functions are uniquely determined by the multi–slit . Hence Theorem 1.1 provides a canonical way of describing a multi–slit by a growth process modeled via a Loewner–type equation. We therefore call the differential equation (1.2), i.e., the multi–slit chordal Loewner equation with constant weights, the Schramm–Loewner equation for the multi–slit .
Remark 1.3 (The multi–slit Loewner equation in Mathematical Physics)
We note in passing that the multiple–slit equation (1.3) has recently been used in mathematical physics for the study of certain two–dimensional growth phenomena. For instance, in [CM02] the authors analyze “Laplacian path models”, i.e. Laplacian growth models for multi–slits. By mapping the upper half–plane conformally onto a half-strip one obtains a Loewner equation for the growth of slits in a half–strip, which can be used to describe Laplacian growth in the “channel geometry”, see [GS08] and [DV11]. Furthermore, equation (1.3) can be used to model so–called multiple Schramm–Loewner evolutions, see [KL07] and [Car03], [BBK05], [Dub07], [Gra07].
Remark 1.4 (The multi–finger radial Loewner equation; Prokhorov’s theorem)
For the radial Loewner equation on the unit disk , the multi–slit situation has already been studied long time ago by Peschl [Pes36] in 1936. He proved that for every union of Jordan arcs in such that is simply connected, there are continuous weight functions with and for every , and continuous driving functions such that the solution to the radial Loewner equation
has the property that maps conformally onto . As in the chordal case, this representation of the multi–slit is not unique. However, if the Jordan arcs are piecewise analytic, it is has been proved by D. Prokhorov [Pro93, Theorem 1 & 2] that one can choose constant weight functions and that then these weights as well as the continuous driving functions are uniquely determined. Prokhorov’s result forms the basis for his original and penetrating control–theoretic study of extremal problems for univalent functions, see his monograph [Pro93]. Clearly, Prokhorov’s result is the analogue of Theorem 1.1 for the radial Loewner equation (1.4), but only under the very restrictive additional assumption that the multi–slit is piecewise analytic. An extension of Prokhorov’s theorem for not necessarily piecewise analytic slits, i.e., the full analogue of Theorem 1.1 for the radial case will be discussed in the forthcoming paper [BS].
Remark 1.5 (Schramm–Loewner constants)
We call the constant weights in Theorem 1.1 the Schramm–Loewner constants of the multi–slit . Is there a interpretation for the Schramm–Loewner constants in terms of geometric or potential theoretic properties of ? Since our proof of Theorem 1.1 is non–constructive, it would be interesting to find a method for computing the Schramm–Loewner constants for a given multi–slit .
We will now outline the main idea of the proof of Theorem 1.1 (Existence) for the case of a two–slit . Roughly speaking, we use a “Bang–Bang Method” based on the one–slit Loewner equation (1.1). Let and be two slits with disjoint closures. We can assume . By extending and , we can find two slits and with disjoint closures such that .
Step 1: Let be a step function. We construct two continuous driving functions such that the solution to the Loewner equation
at time satisfies , where the two–slit is a subset of . Informally, the two–slit is generated by letting grow whenever , and by letting grow whenever . Note that (1.5) has the form of the one–slit Loewner equation (1.1) but with a discontinuous (“bang–bang”) driving function.
Step 2: We show that the set of all driving functions from Step 1 is a precompact subset of the Banach space of continuous functions on equipped with the sup–norm . The proof of this key observation requires a fair amount of technical work, which will be carried out in Section 2 and Section 3.
Step 3: We construct a sequence of step functions such that:
For every the two–slit generated by the step function via Step 1 is exactly the two–slit .
The sequence converges weakly in the Banach space to a constant .
Each step function is constructed as follows. We divide into disjoint intervals of equal length and let . On each of these intervals we let grow on the first subinterval of length and we let grow on the remaining subinterval of length . A continuity argument shows that there is a number such that this process generates exactly the two–slit . Passing to a subsequence if necessary, we may assume that is convergent with limit . The corresponding step functions then do have the required properties (i) and (ii).
Step 4: Using the step functions of Step 3, we construct the corresponding driving functions and by Step 1. With the help of Step 2, we get subsequential limit functions and finally show that the solution to
has the property that .
This paper is organized as follows. In Sections 2 and 3 we provide a number of technical, but crucial auxiliary results, which will be used in Section 4 for the proof of the existence statement of Theorem 1.1. In Section 5 we establish a dynamic interpretation of the weights , which will be employed for the proof of the uniqueness statement of Theorem 1.1 in Section 6. We shall give the details only in the case , i.e., for two slits. The general case of slits can be proved in exactly the same way by induction.
2 The two–slit chordal Loewner equation
We first recall that a bounded subset is called a hull if and is simply connected, so every slit and every multi–slit is a hull. For a hull we denote by the unique conformal mapping from onto such that
where is the half–plane capacity of .
Now, let and be slits such that is a hull. We call a pair of continuous functions with , , a Loewner parametrization for the hull , if the following two conditions hold:
Both functions, and , are nondecreasing;
for every .
Informally, , , is a continuously increasing family of subhulls of such that for every at least one of the two slits is growing. The functions
are called the weight functions of the Loewner parametrization . Note that are well defined for a.e. as derivatives of nondecreasing functions and they belong to the space of –functions on the interval . Moreover, and for a.e. by (ii). Informally, and measure the speed of growth of and w.r.t. half–plane capacity. If we let , then the functions
are called the driving functions of the Loewner parametrization . As in the one–slit case, the driving functions are continuous (see also Theorem 2.2).
If is a Loewner parametrization, then the evolution of the family of subhulls can be described by the two–slit chordal Loewner equation as follows.
Remark 2.1 (The two–slit chordal Loewner equation)
Let be slits such that is a hull and let be a Loewner parametrization of with weight functions and driving functions . Then the conformal map is the solution of the Loewner equation
A proof of Remark 2.1 can be given along the lines of the proof of Theorem A in [GM13]. We do not give the details here mainly because we need the statement of Remark 2.1 only in a very special case, which can be deduced fairly quickly from the one–slit Loewner equation (see Lemma 4.1 below). In particular, the proof of Theorem 1.1 does not depend on Remark 2.1, but only on Theorem A.
Note that, in view of Remark 2.1, for proving the existence part of Theorem 1.1, we essentially have to show that every two–slit with has a Loewner parametrization with constant weight functions. To this end, we arbitrarily choose two slits and with disjoint closures such that , and consider all possible Loewner paramaterizations of subhulls of . The key result is then the following theorem.
Let be slits with disjoint closures and . Then the set of driving functions for all Loewner parametrizations of subhulls of is a compact subset of the Banach space .
The proof of Theorem 2.2 is divided into two parts. In this section, we show that the driving functions in Theorem 2.2 form a closed subset of , and we defer the more difficult proof of precompactness to Section 3.
Let with and for a.e. , and let . For every let be the supremum of all such that the solution of the initial value problem (2.1) is well defined up to time with . Let . Then is the unique conformal map from onto such that as .
We call the Loewner chain associated to the weight functions and the driving functions . The following result shows that the Loewner chain depends continuously on its weight functions and its driving functions, provided we choose the appropriate topologies. Recall that a sequence of Loewner chains is said to converge to the Loewner chain with domain in the Carathéodory sense, if for every , converges to uniformly on , where is the closure of , see [Law05, §4.7].
Theorem 2.4 (Continuous dependence of Loewner chains)
For let be weight functions and let be driving functions with associated Loewner chains , . If converges to uniformly on and if converges weakly in to for , then converges in the Carathéodory sense to the chain .
Theorem 2.4 generalizes Proposition 4.47 in [Law05], which deals with the one–slit version of Loewner’s equation. The idea of the statement and the proof of Theorem 2.4 comes from a standard result in linear control theory (see [Jur97, p. 117]) by thinking of the weight functions as “control functions”.
Proof of Theorem 2.4.
For every , let
Then is an open subset of . As in [Law05, p. 115], it suffices to show that converges to uniformly on .
We first need to establish a number of technical, but crucial estimates.
Since converges weakly to , we have as pointwise on . In fact, this convergence is uniform on , since the sequence is equicontinuous there. This follows from
for all and .
so as by assumption. Let with . Then there is a positive integer such that for all . Since uniformly on by (i), we may assume by enlarging if necessary that
for all and all .
(iii) Let and let be the first time such that . If , then for all ,
We are now in a position to show that converges to uniformly on . Let . Then
Therefore, we have for all and every in view of of (ii) and (iii),
The Gronwall lemma [FR75, p. 198] shows that this estimate implies
Hence, in view of (2.2), we get for all and every . In particular, , so we have for all
This completes the proof of Theorem 2.4. ∎
Proof of Theorem 2.2 I: Closedness.
Let be driving functions for Loewner parametrizations of subhulls of , and assume that uniformly on for . Let be the corresponding weight functions and the associated Loewner chains. As the set of functions in with values (a.e.) in the interval is a weakly compact subset of , we can assume that converges weakly to some , where and for a.e. . Let be the Loewner chain associated to and . By Theorem 2.4, in the Carathéodory sense. Let and be the domains of and . Since the sets are subhulls of , also is a subhull of , so , where . Clearly, is a Loewner parametrization of the subhull of with driving functions . ∎
3 Capacity estimates and proof of Theorem 2.2.
Let be slits with disjoint closures and . In this section we will finish the proof of Theorem 2.2 by showing that the set of driving functions for all Loewner parametrizations of subhulls of is a precompact subset of the Banach space . This requires a number of technical estimates for the half–plane capacities of two–slits and their subhulls.
We start with the following lemma, which describes a number of well–known, but essential properties of half–plane capacity. For a geometric interpretation of half–plane capacity, we refer to [LLN09, RW].
Let be hulls.
If and are hulls, then
If is a hull and , then
For (a) and (b) see [Law05, p. 71]. Now let be a hull such that . Then (b) implies , while (a) shows . This proves (c). ∎
Next we prove a refinement of Lemma 3.1 (c) when the hulls are slits.
Let and be slits with disjoint closures. Then there is a constant such that
for all subslits .
We note that a local version of Lemma 3.2 in the sense of
Using for and Lemma 3.1, it is easy to see that
Let and let be the parametrization of by its half–plane capacity (see [Law05, Remark 4.5]). For fixed let and . Hence, in view of (3.1) and since , all we need to show is that there is a constant such that
In order to prove (3.2), we proceed in several steps.
(i) Fix . For let
Then, by [LSW01, Lemma 2.8], the right derivatives and of and at exist and
where the limit is taken over . Now note that by Lemma 3.1 b),
and, in a similar way, . Therefore, (3.3) shows that the right derivative of the function exists for every and
(ii) Next is continuous on (see [Law05, Lemma 4.2]). Furthermore, since , i.e., , the function has an analytic continuation to a neighborhood of and , see [Law05, p. 69]. Since is continuous in the topology of locally uniform convergence, we hence conclude from (3.4) that is a continuous nonvanishing function on the interval .
(iii) From (ii) we see that