Non-Slit Solutions to the Loewner Equation

Abstract

We study conditions on the driving term of the Loewner equation which guarantee that the equation generates slits. We construct a new family of examples of non-slit solutions which generalize Kufarev's example, in which the local Lipschitz characteristic of the driving term takes values from approx. 4.01 to plus infinity. This should be compared to results by Marshall, Rohde and Lind, which state that driving terms with local Lipschitz characteristic less than 4 always generate slit solution.