Exact solutions of Loewner equations
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Exact solutions of Loewner equations Hai-Hua Wu1,2 Received: 25 January 2018 / Revised: 25 January 2018 / Accepted: 12 October 2020 © Springer Nature Switzerland AG 2020
Abstract We consider the exact singular solution of chordal Loewner equation and investigate a sequence of vertical slits γ p , where p = 4n + 1, n ∈ N. Our main result is to give an exact expression of the driving function λ, and its Hölder exponent near 0 in terms of p, which lies in (1/2, 1] and has a natural connection with the known results. Keywords Loewner equation · Hull · Half-plane capacity · Driving function · Trace Mathematics Subject Classification Primary 30C30; Secondary 30C45
1 Introduction The Loewner differential equation was introduced by Loewner in 1923 to study the Bieberbach conjecture, and was the main tool in the final solution of the conjecture by de Branges [1]. In 2000, Schramm [11] randomized the equation to the so-called stochastic Loewner evolutions (SLE) by taking Brownian motions as the driving functions. With the work of Lawler, Rohde, Werner, Smirnov and many others, SLE has grown to be an important research topic in probability and conformal field theory. This also re-ignited the interest of the equation and its solution in the deterministic case. Let H be the upper half-plane. Suppose for any T > 0, γ : [0, T ] → H is a simple curve with γ (0) ∈ R and γ (0, T ] ⊂ H. For each t ∈ [0, T ], the region Ht = H\γ [0, t] is a simply connected subdomain of H. There is a unique conformal map gt from Ht
The research is supported in part by the NNSF of China (No. 11701166).
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Hai-Hua Wu [email protected]
1
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, People’s Republic of China
2
School of Mathematics, Hunan University, Changsha 410082, People’s Republic of China 0123456789().: V,-vol
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H.-H. Wu
onto H such that gt (z) = z +
b(t) +O z
1 |z|2
, as z → ∞.
If we change the parameterization of γ such that b(t) = 2t, then γ is said to be parameterized by half-plane capacity. In this case, gt (z) satisfies the equation ∂ 2 gt (z) = , g0 (z) = z, ∂t gt (z) − λ(t)
(1.1)
where λ(t) := lim z→γ (t) gt (z) is a continuous real-valued function. The equation (1.1) is called (chordal) Loewner (differential) equation, and gt are called Loewner chains. λ is called the driving function or the Loewner transform, and γ is called the trace or the Loewner curve. On the other hand, given a continuous function λ : [0, T ] → R and z ∈ H, we can solve the initial value problem (1.1). Let Tz be the supremum of all t such that the solution is well defined up to time t with gt (z) ∈ H. Let Ht := {z ∈ H : Tz > t} . Then gt is the unique conformal transformation from Ht onto H with 2t gt (z) = z + +O z
1 |z|2
, as z → ∞.
(1.2)
Let K t := H \ Ht . Then {K t }t∈[0,T ] is an increasing family of hulls (defined in Sect. 2), and we can say that the hulls K t are generated by the driving function λ. In general, it is not true that {K t }t∈[0,T ] generated by
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