María J. Martín Gomez: A new definition of the Schwarzian derivative for harmonic happings

A complex-valued harmonic function in a simply connected domain has a canonical representation $f=h+\overline g$ (where $h$ and $g$ are analytic functions) that is unique up to an additive constant.
Any harmonic mapping $f$ with $|h^\prime|+|g^\prime|\neq 0$ lifts to a mapping $\widetilde{f}$ onto a minimal surface defined by conformal parameters if and only if the (second complex) dilatation $\omega=g^\prime/h^\prime$ equals the square of an analytic function $q$. In other words, $\omega=q^2$ for some analytic function $q$. M.
Chuaqui, P. Duren, and G. Osgood presented in 2004 a definition of the Schwarzian derivative for this kind of harmonic mappings.

In a joint work with R. Hernández, we introduce a new definition of the Schwarzian derivative for any locally univalent harmonic mapping in the complex plane $f$ without assuming any additional condition on the dilatation of $f$.