Duality and singularities of worldsheet along space-like framed curves in de Sitter 3-space

This paper generalizes the construction of worldsheets generated by space-like curves, including both regular and singular cases, in de Sitter 3-space [Formula: see text]. Building on a Lorentzian Frenet–Serret formalism for framed curves with curvature invariants [Formula: see text], we present a unified two-case parametrization accommodating singularities. The generalized worldsheet [Formula: see text] admits the parametrization :[Formula: see text]The de Sitter brane singular curve [Formula: see text] encodes singular loci via curvature-driven transitions. Algebraic conditions on curvature functions classify nondegenerate singularities (cuspidal edges and swallowtails) using unfolding theory in singularity theory and the differential geometry of fronts. By unifying regular and singular regimes through curvature-driven kinematics, this work provides new tools for studying degeneracy propagation in Lorentzian geometry, particularly transitions in causal structures and bifurcations of singular loci, as demonstrated through physical examples modeling black hole mergers and cosmological horizons.