Polygonal negative hyperbolic rotopulsators of the curved $n$-body problem

Abstract

For the $n$-body problem in spaces of negative constant Gaussian curvature, we prove for a class of negative hyperbolic rotopulsators that if that class exists, the configurations of the point masses of these rotopulsators have to be regular polygons if the rotopulsators are not relative equilibria. Additionally, we prove that if the rotopulsators are relative equilibria, there exists at most one such solution.