Behavior of Gauss curvatures and mean curvatures of Lightcone framed surfaces in the Lorentz-Minkowski 3-space
Chang Xu, Liang Chen
TL;DR
The paper investigates Gaussian and mean curvatures of lightcone framed surfaces in Lorentz-Minkowski $3$-space, focusing on curvature behavior at lightlike and singular points of mixed-type surfaces. It introduces a modified frame to extend curvature definitions to these singular regimes and derives explicit expressions for modified Gauss and mean curvatures, along with modified principal curvatures and associated invariants. Key results establish when curvature quantities remain bounded or diverge near lightlike points and 1st singular points, and relate these behaviors to the underlying kind of singularity. An explicit example verifies the theoretical framework, illustrating the lightlike and singular loci and the unboundedness of curvatures at lightlike points. These findings advance the differential-geometric analysis of mixed-type surfaces by providing robust curvature notions across singular regimes and clarifying the impact of lightlike geometry on curvature behavior.
Abstract
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 behavior of the Gaussian curvature and mean curvature of the lightcone framed surface at not only lightlike points but also singular points.