Cuspidal edges and generalized cuspidal edges in the Lorentz-Minkowski 3-space

It is well-known that every cuspidal edge in the Euclidean space E³ cannot have a bounded mean curvature function. On the other hand, in the Lorentz-Minkowski space L³, zero mean curvature surfaces admit cuspidal edges. One natural question is to ask when a cuspidal edge has bounded mean curvature in L³. We show that such a phenomenon occurs only when the image of the singular set is a light-like curve in L³. Moreover, we also investigate the behavior of principal curvatures in this case as well as other possible cases. In this paper, almost all calculations are given for generalized cuspidal edges as well as for cuspidal edges. We define the "order" at each generalized cuspidal edge singular point is introduced. As nice classes of zero-mean curvature surfaces in L³, "maxfaces" and "minfaces" are known, and generalized cuspidal edge singular points on maxfaces and minfaces are of order four. One of the important results is that the generalized cuspidal edges of order four exhibit a quite similar behaviors as those on maxfaces and minfaces.