Singular and tangent slit solutions to the Loewner equation

Abstract

We consider the Loewner differential equation generating univalent maps of the unit disk (or of the upper half-plane) onto itself minus a single slit. We prove that the circular slits, tangent to the real axis are generated by H\"older continuous driving terms with exponent 1/3 in the Loewner equation. Singular solutions are described, and the critical value of the norm of driving terms generating quasisymmetric slits in the disk is obtained.