Behaviors of Gauss curvatures and mean curvatures of Lightcone framed surfaces in the Lorentz-Minkowski 3-space

In this paper, we investigate the differential geometric properties of lightcone framed surfaces in Lorentz-Minkowski 3-space. In general, a mixed type surface is a connected regular surface with non-empty spacelike and timelike point sets. While a lightcone framed surface is a mixed type surface with singular points at least locally. We introduce a useful tool, so called modified frame along the lightcone framed surface, to study the differential geometric properties of the lightcone framed surface. As results, we show the behaviors of the Gaussian curvature and mean curvature of the lightcone framed surface at not only lightlike points but also singular points.