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In this talk which is based on the joint work with Kentaro Saji given in [1], taking into account the Legendrian dualities in [2] which are extensions of the Legendrian dualities in [3], we first introduce new extended Legendrian dualities for the $3$-dimensional pseudo-spheres of various radii in Lorentz-Minkowski $4$-space. Secondly, by connecting all of these Legendrian dualities continuously, we construct Legendrian dual surfaces (lying in these $3$-dimensional pseudo-spheres) of a spacelike curve in the $3$-dimensional lightcone. Finally, we investigate the singularities of these surfaces and show the dualities of the singularities of a certain class of such a surface in the $3$-dimensional lightcone.

1. S. Izumiya, "Legendrian dualities and spacelike hypersurfaces in the lightcone", Moscow Mathematical Journal, 9 (2009), 325-357.

2. S. Izumiya, H. Yıldırım, "Extensions of the mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space", Topology and its Applications, 159 (2012), 509-518.

3. K. Saji, H. Yıldırım, "Legendrian dual surfaces of a spacelike curve in the 3-dimensional lightcone", Journal of Geometry and Physics, 179 (2022) 104593, 1-21.