<abstract><p>In the realm of the four-dimensional Minkowski space $ \mathbb{L}^{4} $, the focus is on hypersurfaces classified as right conoids and defined by light-like axes. Matrices associated with the fundamental form, Gauss map, and shape operator, all specifically tailored for these hypersurfaces, are currently undergoing computation. The intrinsic curvatures of these hypersurfaces are determined using the Cayley-Hamilton theorem. The conditions of minimality are addressed by the analysis. The Laplace-Beltrami operator for such hypersurfaces is computed, accompanied by illustrative examples aimed at fostering a more profound understanding of the involved mathematical principles. Additionally, scrutiny is applied to the umbilical condition, and the introduction of the Willmore functional for these hypersurfaces is presented.</p></abstract>