Vortex solutions of Liouville equation and quasi spherical surfaces

Abstract

We identify the two-dimensional surfaces corresponding to certain solutions of the Liouville equation of importance for mathematical physics, the nontopological Chern–Simons (or Jackiw–Pi) vortex solutions, characterized by an integer [Formula: see text]. Such surfaces, that we call [Formula: see text], have positive constant Gaussian curvature, [Formula: see text], but are spheres only when [Formula: see text]. They have edges, and, for any fixed [Formula: see text], have maximal radius [Formula: see text] that we find here to be [Formula: see text]. If such surfaces are constructed in a laboratory by using graphene (or any other Dirac material), our findings could be of interest to realize table-top Dirac massless excitations on nontrivial backgrounds. We also briefly discuss the type of three-dimensional spacetimes obtained as the product [Formula: see text].