Hyperbolic Shirts fit a 4-body Problem

Consider the equal mass planar $4$-body problem with a potential corresponding to an inverse $\mathit{\text{cube}}$ force. The Jacobi-Maupertuis principle reparametrizes the dynamics as geodesics of a certain metric. We examine the curvature of this geodesic flow in the reduced space on the collinear and parallelogram invariant surfaces and derive some dynamical consequences. This proves a numerical conjecture of [4].