Classification of flat slant surfaces in complex Euclidean plane

It is well-known that the classification of flat surfaces in Euclidean 3space is one of the most basic results in differential geometry. For surfaces in the complex Euclidean plane C2 endowed with almost complex structure J, flat surfaces are the simplest ones from intrinsic point of views. On the other hand, from J-action point of views, the most natural surfaces in C2 are slant surfaces, i.e., surfaces with constant Wintinger angle. In this paper the author completely classifies flat slant surfaces in C2. The main result states that, beside the totally geodesic ones, there are five large classes of flat slant surfaces in C2. Conversely, every non-totally geodesic flat slant surfaces in C2 is locally a surface given by these five classes.