The purpose of this paper is to interpret the phase transition in the Loewner theory as an analog of the hyperbolic variant of the Schur theorem about curves of bounded curvature. We define a family of curves that have a certain conformal self-similarity property. They are characterized by a deterministic version of the domain Markov property, and have constant Loewner curvature. We show that every sufficiently smooth curve in a simply connected plane domain has a best-approximating curve of constant Loewner curvature, establish a geometric comparison principle, and show that curves of Loewner curvature bounded by 8 are simple curves.
1 Introduction and Results
A classical theorem of A. Schur [S] says that, among all curves in the plane of curvature bounded above by , the circular arc of (constant) curvature K minimizes the distance between the endpoints (assuming the length of the curve is smaller than ). The analog for the hyperbolic plane was established in [FG]: Again, curves of constant (geodesic) curvature are extremal, but an interesting phenomenon occurs. Namely, if the geodesic curvature (which can simply be defined at the point 0 in the hyperbolic unit disc as half of the euclidean curvature, and then at arbitrary points via isometry) is , then the curve is automatically a simple curve that tends to the boundary of , whereas the curves of constant curvature are circles in in thus do not tend to the boundary.
A similar “phase transition” occurs in the Loewner theory: In the framework of the chordal Loewner equation in the upper half plane (see Section 3.1), if a curve has driving term which is Hölder continuous with exponent and norm , then is a simple curve, whereas for norm the curve can have self-intersections, see [Li] and [RTZ].
The purpose of this paper is to define a quantity that we call Loewner curvature, and to establish some of its properties, particularly regarding the above phenomenon. Our geometric definition of Loewner curvature will depend on a family of curves, introduced in Section 2, that we call self-similar. They are motivated by the Schramm-principle and will be used as comparison curves. In particular, these are our curves of constant curvature. In Section 3, we identify these curves using the Loewner differential equation, and obtain that it is a two-parameter family of curves. A one-dimensional collection of these curves is depicted in Figure 1. We also compute the first terms of the series expansion of a curve in terms of the first and second derivative of the Loewner driving term, and show in Proposition 3.3 that
for near 0, where and . This allows us (Proposition 3.4) to associate with every sufficiently smooth curve a best-fitting comparison curve. With this foundation, we can make our geometric definition for Loewner curvature and obtain our Loewner curvature formula
in Section 4. There we also show (Theorem 4.4) that is a simple curve if . The constant is best possible and corresponds to the constant 4 in the criterion for simple Loewner traces. In Section 5, we establish a comparison principle (Theorem 5.1) by showing that a bound on Loewner curvature implies that the curve stays to one side of the corresponding curve of constant curvature.
Acknowledgement: We thank Fredrik Viklund for his comments on the first version of the manuscript.
2 A family of curves
In this section, we will introduce a family of curves, which we call “self-similar curves.” In Section 4, we will define the Loewner curvature for these curves to be constant. The Loewner curvature for an arbitrary curve will be defined by comparison to this family. In order to give some motivation for our definition of the self-similar curves, we will first remind the reader of Schramm’s principle and then formulate a deterministic version.
Let be a Jordan domain with two distinct marked boundary points, and . For each triple , we assume that there is a family of simple curves in that begin at and a probability measure on . (Note: This could be made slightly more general: we do not need simple curves, but we must require that the curves do not cross over themselves.) The measures are said to satisfy Schramm’s principle if they satisfy the following:
Conformal invariance: Given a conformal map of the domain , then
(The measure on the family of curves is the push-forward of the measure .)
Domain Markov property: Let be the initial part of a curve in . Let be the conditional probability measure given . Then
or in other words, given the initial part of the curve , the conditional measure is the same as the measure on curves starting at in the slit domain .
Given , define to be the set of all conformal maps with and . Thus if satisfy conformal invariance and the domain Markov property, then
for any . This means that the measure is invariant under “mapping down” the initial part of a curve. Measures that satisfy Schramm’s principle must be from the one-parameter family of SLE measures.
This leads us to a deterministic Schramm’s principle, that is, a deterministic axiom that will characterize an important family of curves. In this paper we will be interested in curves in a domain with two marked boundary points. Let us introduce our notation:
Given a Jordan domain with distinct boundary points and and given , the notation means that is a simple (deterministic) curve with and .
We have several comments about the definition: First, the point does not enter into the definition, and in particular, we do not require that . However, we think of as being the location of an observer. When we introduce our notion of Loewner curvature later, it will be relative to this observer. Second, we note that when is defined, it is possible that or . Next, we consider two curves equivalent if one is a reparametrization of the other. For the most part, we are not concerned with the particular parametrization of a curve; however when working in the Loewner framework, we will use the typical halfplane-capacity parametrization. Lastly, for ease of notation, we will often simply write for , the image of the open time interval.
We want to understand the curves that satisfy the following self-similarity property:
A curve is self-similar if and if for each , there exists a conformal map so that
We write for the family of self-similar curves in the marked domain .
In other words, self-similar curves are invariant under “mapping down” any initial part of the curve, modulo a conformal renormalization. In Theorem 3.2 below, we will determine all self-similar curves. In particular, we obtain the following:
Given a Jordan domain with distinct boundary points and , there exists a unique two-parameter family of self-similar curves.
3 Self-similar curves and the Loewner framework
If we know for one fixed triple , then we can obtain via a conformal map with and . That is,
It is most convenient to work with , the upper half-plane, with and , and for the remainder of this section we will work in this setting. We take the viewpoint that doing computations with the triple is like “working in coordinates.”
Our goal is to use the chordal Loewner equation to understand the family . We begin with a brief review of the chordal Loewner equation (that the expert can safely skip.)
3.1 The chordal Loewner equation
Let be continuous, and let . Then the chordal Loewner differential equation is the following initial value problem:
We call the driving function, since it determines the unique solution , which is guaranteed to exist for some time interval. The only obstacle in solving (3.1) is obtaining a zero in the denominator. We collect these problem points together in the set , called the (Loewner) hull. So
If , then is well-defined, and it can be shown that is simply connected and is the unique conformal map from onto that satisfies the following normalization at infinity (called the hydrodynamic normalization):
Further, , and this quantity is called the halfplane capacity of . For more details, see [La].
In the simplest situation, there is a simple curve so that . It is also possible that is the complement of the unbounded component of for a non-simple curve that touches itself or . In both these situations, the curve is called the trace. For example, if , then is the simple curve shown on the left of Figure 2. When , then is the union of the curve shown in the right of Figure 2 and the region under the curve. See [KNK] for a detailed discussion of these examples. There are other possibilities for (such as a space-filling curve), but these do not concern us in this paper.
The Loewner equation provides a correspondence between continuous driving functions and certain families of hulls. Above, we briefly explained how one can start with a continuous function and use the Loewner equation to obtain a hull. On the other hand, if we have an appropriate family of hulls333For the precise statement, see Section 4.1 in [La]., we can determine the driving function. We will assume for simplicity that the family of hulls is for a simple curve in with . Let be the conformal map with the hydrodynamic normalization at infinity (the normalization given in (3.2).) We can reparameterize as necessary so that in (3.2) , in which case we say that is parametrized by halfplane capacity. Then one can show that the conformal maps satisfy (3.1) with , that is, is the conformal image of the tip of the curve .
We list some simple but useful properties of the chordal Loewner equation. Assume that the hulls are generated by the driving term .
Scaling: For , the driving term of the scaled hulls is .
Translation: For , the driving term of shifted hulls is .
Reflection: The driving term of the reflected hulls is , where denotes reflection in the imaginary axis.
Concatenation: For fixed , is the driving function of the mapped hulls .
Theorem 3.1 (Earle, Epstein, Marshall).
Assume has driving function . If any parametrization of is , then the halfplane-capacity parametrization of is in and , where is the halfplane-capacity of (and may be infinite.)
3.2 Computations in coordinates
The following theorem implies Corollary 2.3.
Assume has driving function . If , then is completely determined by the two real parameters and . In fact, if and only if is one of following driving functions:
where and .
Before the proof, we wish to briefly describe the curves generated by the driving functions listed in (3.3).
If , then will be the ray .
If , then will be a scaled version of the left curve shown in Figure 3 (and reflected around the imaginary axis if ). This curve goes to infinity, but remains within a bounded distance of the real line.
If and , then will end with an infinite spiral. The spiral is so tight as to not be visually discernable, as in the example shown in Figure 2. If and , then , and the angle of intersection depends on . These examples were first described and implicitely computed in [KNK]. In both cases, changing scales the picture.
If , then will approach infinity asymptotic to a ray, as shown in the right-hand example in Figure 3. The angle of the ray depends on , and changing scales the picture.
Remark: The notion of self-similarity introduced in this paper is a generalization of the notion introduced in [LMR], where we defined a curve in with finite halfplane capacity to be self-similar if is a translation and dilation of for all . Proposition 3.1 in [LMR] states that is self-similar (under the more restrictive definition) if and only if the driving function is for some and .
[LMR] also contains geometric constructions for these curves, which gives a conceptual and simple way to obtain
the solutions to with driving function .
Now for the proof of Theorem 3.2:
Assume that . Without loss of generality, we assume is parametrized by halfplane capacity. This means that for each , there exists a conformal map with and so that It follows that , where and is the solution to (3.1). Thus
By the concatenation, scaling and translation properties,
If , then for all , and hence . If , then for all and for . If and , then and are nonzero for all , and (3.5) gives
Setting and solving (3.6) gives
Thus, there exists and so that (when ) or (when .)
Conversely, if is one of the driving functions or then will satisfy (3.4), and this implies that . ∎
Assume that is the driving function for a curve . Further, we assume that and either or both and are 0. Then in the halfplane-capacity parametrization, satisfies
for near 0, where and .
This proof needs two components: computations and Theorem 3.3 in [W]. For the computations, we need to show that the curves generated by driving functions and all satisfy (3.7). These computations are straightforward. As done in [KNK], one can solve (3.1) for the driving functions in our list, obtaining an implicit equation for . Then we simply expand in a power series to obtain (3.7).
For the second step, choose to be the driving function in this list that satisfies and . Let be the curve generated by . Then Theorem 3.3 in [W] implies that for ,
for small enough . This in turn implies (3.7).
The previous proposition shows that the two parameters and determine the Loewner trace up to 4th order in (excluding the case and .) These two parameters also uniquely determine a self-similar curve in by Theorem 3.2. As we see in the next proposition, this allows us to match a given curve with a unique “best-fitting” self-similar curve. We say is the best-fitting curve to at if under the conformal transformation the halfplane-capacity parametrization of and match up to 4th order (or satisfy .)
Let be . At each point , the curve has a unique best-fitting curve .
By conformal transformation, it suffices to consider the situation when , and . By Theorem 3.1, since is , we can deduce that both and the halfplane-capacity parametrization of are in (where is the halfplane capacity of and may be infinite.) For the rest of the proof, we assume is parametrized by halfplane capacity and .
For , and are defined. If , we set . Otherwise, we choose the unique satisfying and . Then by Proposition 3.3 and Theorem 3.2, is the unique best-fitting curve to at the origin. Thus is the unique best-fitting curve to at the point .
4 Loewner curvature
4.1 The definition of Loewner curvature
We are now ready to give the definition of Loewner curvature, notated . We begin with assigning constant curvature for as follows:
Let . Then , the Loewner curvature of , is defined to be the following constant:
If is generated by , then .
If is generated by , then .
If is generated by , then .
If is generated by , then .
Note that for , the Loewner curvature is scaling invariant by definition. For the self-similar curves , is defined to satisfy conformal invariance. For a curve , we define by comparison to the curves of constant curvature as follows:
Let be . The Loewner curvature of at the point , notated , is defined to be , where is the unique best-fitting curve to at .
Although our approach to defining Loewner curvature via comparison curves is natural, it is difficult to use this definition to compute . The following lemma shows that computations are straightforward in the Loewner framework. (The nice formula that appears in this lemma also explains the particular definition for in cases 3 and 4 of Definition 4.1.)
Let be a curve that is parametrized by halfplane capacity, and let be the corresponding driving function. Then for ,
Since precisely when , if the right-hand side of (4.1) is in the indeterminate form , we declare it to equal 0.
The last equality is due to Proposition 3.3.
The next theorem says that if the curvature is small, then cannot “curve” enough to hit itself or the boundary (except at the marked point .)
Let be . If for all , then is a simple curve in .
The constant 8 is best possible, as the curve in case 3 of Definition 4.1 with shows. The following lemma is key component of the proof.
Let be with driving function . If , then is Lip with norm bounded above by .
Set , and assume that (if not, replace with .) Then
The solution to is
for some . Assume for the moment that , and let . Then
It remains to show for all . Suppose to the contrary that there is a time so that . This will imply, as we show in the remainder of the proof, that there is a time with , contradicting the fact that .
For simplicity, assume (by shifting time if necessary), and set
Let be large enough so that
This is possible since and the limit as of the left hand side is . Recursively define an increasing sequence by
Finally set , which satisfies
Now the proof of Theorem 4.4:
We split this proof into two cases. For the first case, suppose that there exists a time so that . We claim that for all . If this claim is not true, then there must be a first time so that . This implies that either or . However, for , and must have the same sign, implying that . Hence, and are bounded away from zero on , proving the claim. Now in this case, Lemma 4.5 implies that is Lip on with norm strictly less than 4. Thus by Theorem 2 in [Li], is a simple curve in . Therefore is simple in .
For the next case, suppose that for all . This means that and always have the opposite sign (or could be 0). Thus, for , is bounded on , which implies that is locally Lip with small norm. In particular, we can find an interval so that is Lip on with norm strictly less than 4. We finish the proof as in the first case.
For , Loewner curvature is invariant under scaling. This follows from Definitions 4.1 and 4.2 and can also be verified computationally by Proposition 4.3. To distinguish between different curves with the same Loewner curvature, we must choose a “scale” which we define as follows:
Assume has driving function with . Then the scale of at 0 is , which is the coefficient of the linear term in (3.7).
The scale of at 0 is simply a multiple of the Euclidean curvature of at 0. To see this, let be the upper half-circle with radius , which is driven by . Thus, Proposition 3.3 implies that
Comparing this to (3.7)
shows that the half-circle that best matches the curve at its base has radius .
In other words, the scale of at 0 is twice the Euclidean curvature of at its base.
4.2 Loewner curvature is neither local, nor reversible
Although Loewner curvature shares some similarities with other notions of curvature, there are significant differences. For instance, Loewner curvature is not reversible (but depends on the orientation of the curve), and it is not local (but depends on the “past” of the curve).
To see why this must be true, let’s assume for the moment that Loewner curvature is a purely local concept,
and consider the smooth curve shown in Figure 5. Let be the time that .
If Loewner curvature were purely local, then for , since locally near the curve looks like the vertical slit, which has constant Loewner curvature 0. This means that must be a vertical ray, and therefore would be a hyperbolic geodesic in , which it is not.
This example also shows that Loewner curvature cannot be reversible: in contrast to the forward-direction, if we traverse from to 0, then the Loewner curvature will be 0 prior to reaching .
4.3 Existence of a curve with given Loewner curvature
It is natural to ask the following question:
Question: Given a continuous function ,
does there exist a curve with ?
We will need to revise this question before we can answer it in Theorem 4.7 below. To understand the needed revision, we consider two examples. First suppose is a nonzero constant function. Then there are an infinite number of self-similar curves with Loewner curvature equal to . Thus we should have the freedom to specify another parameter, for instance, the scale (or equivalently, the Euclidian curvature) of the curve at 0. However, if we are in the case that the curve is driven by , then specifying the scale determines the halfplane capacity of the curve. There is a difficulty if is smaller than the desired value of . Since as , it is impossible to continue past so that remains .
For our second example, suppose for , and let be the scale of the curve. Now we wish to solve with . Separation of variables leads to
Regardless of the value of , before time 1. These examples show that the best we can hope for is the existence of a curve on a small time interval.
Let be a continuous function with , and let . Then there exists and a curve so that for and has scale at 0.
Since , there is some interval so that is bounded away from zero. Then we can solve solve
by separation of variables to obtain
By taking a smaller value of if needed, we can ensure that the right-hand side is positive and bounded away from zero for all . Therefore, we can integrate to obtain a function defined on . Now let be the curve generated by via the Loewner equation. Proposition 4.3 implies that , and Proposition 3.3 implies that the scale of at 0 is a.
5 A comparison principle
In Theorem 4.4, a bound on the Loewner curvature yielded global information about a curve. The next theorem is in the same spirit. Let us first set the stage, by introducing the normalized curves of constant curvature, and . For , set
We list some additional terminology and notation that is needed:
For , define , and for , define to be the first time that the tangent vector points downward. We will often use for and for .
The phrase “the base of is to the right of the base of ” means that for and parametrized by height , for small .
The phrase “ is below ” means that the base of is to the right of the base of and the curves and never cross (although we allow them to touch.)
Assume is generated by with and . Let .
If then is below .
If then is below the curve .
If then is below .
If then is below .
The proof of each part of this theorem has two steps. First we must analyze the base of using the power series expansion given in Proposition 3.3. For the second step, we analyze the curve away from its base by comparing the flow under and the flow under (or ). This involves changing time for one of the flows to allow easier comparison.
For and generated by , set
that is, we “map down” by the function and then shift so that the curve is rooted at the origin. We also have the notation and for the corresponding functions and curves associated with the driving function . We wish to compare and . However, we want to compare curves that are the same scale. This leads us to introduce the following time change: for , set
In particular, we will have that .
Assume with and , and let . For , set .
If , then, .
If , then, .
If then .