Asymptotic ratio of harmonic measures of slit sides

The article is devoted to the geometry of solutions to the chordal Loewner equation which is based on the comparison of singular solutions and harmonic measures for the sides of a slit in the upper half-plane generated by a driving term. An asymptotic ratio for harmonic measures of slit sides is found for a slit which is tangential to a straight line under a given angle, and for a slit with high order tangency to a circular arc tangential to the real axis.


Introduction
The famous Löwner differential equation have been introduced in 1923 [7] and was aimed to give a parametric representation of slit domains. In this article we describe an asymptotic behavior of singular solutions and harmonic measures for the sides of a slit in domains generated by a driving term of the Löwner equation.
The chordal version of the Löwner equation deals with the upper half-plane H = {z : Im z > 0}, R = ∂H, and functions f (z, t) normalized near infinity by f (z, t) = z + 2t z + O 1 z 2 which solve the chordal Löwner differential equation (1) df (z, t) dt . The function f (z, t) can be extended continuously to R ∪ γ(t), and f (γ(t), t) = λ(t). The value t is called the half-plane capacity of the curve γ t , t = hcap(γ t ), see, e.g., [6].
We say that γ t ∈ C n , n ∈ N, on [0, S] if, for the arc-length parameter s of γ t , γ(t(s)) has a continuous derivative γ (n) in s on [0, S], t(S) = T . All the derivatives γ (k) , 1 ≤ k ≤ n, at s = 0 are understood as one-side derivatives. A curve γ t ∈ C n , γ(0) = 0, is said to have at least n-order tangency with a ray I θ = {e iθ s : Two curves γ[0, s] ∈ C n and Γ[0, s] ∈ C n are said to have at least n-order tangency at s = 0 if derivatives γ (k) and Γ (k) in s at s = 0 coincide, 0 ≤ k ≤ n.
The extended function f (z, t) maps γ t onto a segment I = is the maximal singular solution to the chordal Löwner equation (1), and f 2 (0, t) is the minimal singular solution to (1). Both of these solutions correspond to the singular point f (0, 0) = 0 of equation (1).
The curve γ t has two sides γ 1t and γ 2t which define different prime ends at the same points, except for its tip. We say that γ 1t is the left side of γ t if going along the boundary of the domain H \ γ t and moving along R from (−∞) to 0, we first meet the side γ 1t and then γ 2t . In this case, γ 2t is called the right side of γ t . The two parts [f 2 (0, t), λ(t)] and [λ(t), f 1 (0, t)] of segment I(t) are the images of the two sides γ 1t and γ 2t of γ t under f (z, t), respectively.
In Section 2, we prove the following theorem.
γ 1t is the left side of γ t , and γ 2t is the right side of γ t .
The most important argument in the proof is the comparison of asymptotic parametric representations of γ t in t and s at s = 0. This approach can be compared with the result by Earle and Epstein [1].
In Section 3, we solve a similar problem for a curve γ t which has at least 6-order tangency with a circular arc in H ∪ {0} tangential to R at the origin. Since a scaling time change t → α 2 t in the Löwner equation (1) is accompanied by changing λ(t) → 1 α λ(α 2 t) and f (z, t) → 1 α f (αz, α 2 t), α ∈ R, we can assume without loss of generality that the circular arc is of radius 1, and the argument of its points is increasing when going from 0. We prove the following theorem. Theorem 2. Let γ t ∈ C 6 , γ(0) = 0, Im γ(t) > 0 and Re γ(t) > 0 for t > 0, have at least 6-order tangency at the origin to the circular arc of radius 1 centered at i, and let f (z, t) map H \ γ t onto H and solve the chordal Löwner equation (1). Then is the left side of γ t , and γ 2t is the right side of γ t .

Proof of Theorem 1
Proof of Theorem 1. For β = 0, Theorem 1 has been proved [10]. The cases β > 0 and β < 0 are symmetric to each other, and we will stop only on β > 0.
The Löwner equation (1) can be integrated in quadratures in particular cases [5].
Suppose that a C 4 -slit γ t satisfies the conditions of Theorem 1. Then there exists a driving function λ(t) ∈ Lip( 1 2 ) such that a solution w = f (z, t) to equation (1) maps H \ γ t onto H. For the arc-length parameter s, γ(t(s)) is represented as (2) γ(t(s)) = e iθ s + o(s 4 ), s → +0. Denote There is c > 0 such that Let s denote the length of γ t(s) , and let σ denote the length of projection of γ t(s) onto I θ(c) (T ) for T large enough. There is a C 4 -dependence s = s(σ), Therefore, Asymptotic expansion (2) implies an asymptotic behavior of a distance between γ t and its projection on I θ , dist(γ t(s) , I θ(c) (σ(s))) = σ + o(σ 4 (s)), s → +0.
Lind, Marshall and Rohde [6] studied the closeness of half-plane capacities for two curves which are close together. According to Lemma 4.10 [6], we have that where σ(τ ) is given by (4). Hence, due to (4) and (5), Take into account (2) and rewrite the last relation in the form Choose an arbitrary sequence {x n } of positive numbers x n , x n → ∞ as n → ∞, and denote We see that and the convergence is uniform with respect to t ∈ [0, T ]. The Radó theorem [11], see also [2, p.60], states that a sequence {h n } of conformal mappings h n from the unit disk D onto simply connected domains D n bounded by Jordan curves ∂D n , 0 ∈ D n , h n (0) = 0, h ′ n (0) > 0, converges uniformly on the closure of D to h : D → D, ∂D is bounded by a Jordan curve, if and only if D n converges to the kernel D and, for every ǫ > 0, there exists N > 0 such that, for all n > N, there is a one-to-one correspondence z n : ∂D n → ∂D, |z n (ζ) − ζ| < ǫ, ζ ∈ ∂D n . Markushevich [8] generalized the Radó theorem to domains with arbitrary boundaries.
Apply the Radó-Markushevich theorem to g n •p with a conformal mapping p from D onto H and obtain that the sequence {g n (w, t)} converges to f −1 c (w, t) as n → ∞ uniformly on compact subsets of H ∪ R.
For k = 1, 2, n ≥ 1, the harmonic measure ω(i; I ′ kn (t), H) of I ′ kn (t) at i with respect to H equals the angle divided over π under which the segment I ′ kn (t) is seen from the point i. Similarly, the harmonic measure ω(i; I kn (t), H) of I kn (t) at i with respect to H equals the angle divided over π under which the segment I kn (t) is seen from the point i, see, e.g., [2, p.334]. This shows that the last term in the chain of equalities has a limit as n → ∞, and This limit exists for every sequence {x n } tending to infinity. So there exists a limit for the ratio of m 1 (t) and m 2 (t) as t → +0, and where β is given by (3). This leads to the conclusion desired in Theorem 1 and completes the proof.

Proof of Theorem 2
Proof of Theorem 2. The Löwner equation (1) admits an explicit integration [9] in the case when γ 0 [0, t] is a circular arc centered at i, γ 0 (0) = 0, with an implicitly given driving function λ(t). To be concrete, we will consider γ 0 [0, t] such that the argument of (γ 0 [0, t] − i) increases in t. Let a solution f 0 (z, t) to equation (1) map t) is represented [9] by the Christoffel-Schwarz integral where β 1 = β 1 (t) and β 2 = β 2 (t) are expanded in powers of 3 √ t, Suppose that a C 4 -slit γ t satisfies the conditions of Theorem 2. Then there exists a driving function λ(t) such that a solution w = f (z, t) to equation (1) maps H \ γ t onto H. For the arc-length parameter s, represent a transformation of γ(t(s)),   (t) ⊂ R and I ′ 2n = I ′ 2n (t) ⊂ R, respectively. Compare I kn (t) and I ′ kn (t) by (12) and conclude that I kn (t) = √ x n I ′ kn (t), and so meas I kn (t) = √ x n meas I ′ kn (t), k = 1, 2, n ≥ 1, 0 < t ≤ T. The harmonic measure is invariant under conformal transformations. This gives that .
For k = 1, 2, n ≥ 1, the harmonic measure ω(i; I ′ kn (t), H) of I ′ kn (t) at i with respect to H equals the angle divided over π under which the segment I ′ kn (t) is seen from the point i. Similarly, the harmonic measure ω(i; I kn (t), H) of I kn (t) at i with respect to H equals the angle divided over π under which the segment I kn (t) is seen from the point i. This shows that the last term in the chain of equalities has a limit as n → ∞, and  (λ 0 (t) − β 1 (t)) 2 β 2 (t) − λ 0 (t) = lim t→+0 (B 1 This leads to the conclusion desired in Theorem 2 and completes the proof.