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

TL;DR

This paper investigates the properties of cuspidal edges in Lorentz-Minkowski space, focusing on conditions for bounded mean curvature and analyzing singularities on zero-mean curvature surfaces such as maxfaces and minfaces.

Contribution

It characterizes when cuspidal edges have bounded mean curvature in L^3 and introduces the concept of order for generalized cuspidal edges, linking them to maxfaces and minfaces.

Findings

Bounded mean curvature occurs only when the singular set maps to a light-like curve.

Generalized cuspidal edges of order four behave similarly on maxfaces and minfaces.

The paper provides detailed calculations for generalized cuspidal edges and their curvature properties.

Abstract

It is well-known that every cuspidal edge in the Euclidean space E^3 cannot have a bounded mean curvature function. On the other hand, in the Lorentz-Minkowski space L^3, zero mean curvature surfaces admit cuspidal edges. One natural question is to ask when a cuspidal edge has bounded mean curvature in L^3. We show that such a phenomenon occurs only when the image of the singular set is a light-like curve in L^3. 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 L^3,"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.