The Loewner and Hadamard variations

Abstract

We give an explicit formula relating the infinitesimal generators of the Loewner differential equation and the Hadamard variation. This is applied to establish an extension of the Hadamard variation to the case of arbitrary simply-connected domains and to prove the existence of Loewner chains with arbitrary smooth initial generator starting at an arbitrary univalent function which is sufficiently smooth up to the boundary. As another application of this method, we show that every subordination chain $f_t$ is differentiable almost everywhere and satisfies a Loewner equation, without assuming that $f_t'(0)$ is continuous.