I was wondering if it is generally possible to have a mapping of a $2$-sphere onto another $2$-sphere for which there are regions of positive and negative gaussian curvature, yet being a minimal surface. $π_2(S_2)=Z$ states that the winding number describes the mapping entirely which suggests that such a mapping is not possible (curvatures of opposite sign will always cancel out). However I do have the additional requirement that all solutions must also suffice laplace's equation (i use laplace's equation to relax the initial mapping to get a minimal surface) and possibly there are mappings which are not reduceable to a single $Z$ under this additional constraint (I'd expect something like two numbers, $+Z/-Z$, a positive and negative winding number in such a case).
Are there known mappings of this kind or how could I get to such solutions ?