Collisions and spirals of Loewner traces

Abstract

We analyze Loewner traces driven by functions asymptotic to κ √ 1 − t. We prove a stability result when κ = 4 and show that κ = 4 can lead to non locally connected hulls. As a consequence, we obtain a driving term λ(t) so that the hulls driven by κλ(t) are generated by a continuous curve for all κ > 0 with κ = 4 but not when κ = 4, so that the space of driving terms with continuous traces is not convex. As a byproduct, we obtain an explicit construction of the traces driven by κ √ 1 − t and a conceptual proof of the corresponding results of Kager, Nienhuis and Kadanoff.

Regularity of Loewner Curves

“…Lind, Marshall and Rohde [18] show that if the driving functions have Hölder-1/2 norm less than 4, then the curves converge uniformly. However, the Brownian motion is a.s. not…”

“…Standard conformal mapping techniques (particularly the GehringHayman inequality) then yield the additional information that the trace is a quasiconformal arc approaching the real line non-tangentially. The constant 4 is sharp as there is a driving function λ with ||λ|| 1/2 = 4 but it does not generate a curve [18].…”

The multiple-slit version of Loewner's differential equation and pointwise Hölder continuity of driving functions

“…which does not generate a simple curve (see [LMR10], chapter 3 and change the time t → λ 2 (1)t). But then, f t (z, d) is not a slit mapping for all d > 0.…”

On tangential slit solution of the Loewner equation

“…The importance of the equation emerged again in the recent study of the stochastic Loewner evolution (SLE) due to Lawler, Schramm and Werner [9,10,11,19,8] and the references there, and Smirnov [20,21,22]. This also re-ignited the interest of the equation and its solution in the deterministic case [2,5,6,12,13,14,15,16,18,24,25].…”