No hyperbolic pants for the 4-body problem

Abstract

The $N$-body problem with a $1/r^2$ potential has, in addition to translation and rotational symmetry, an effective scale symmetry which allows its zero energy flow to be reduced to a geodesic flow on complex projective $N-2$-space, minus a hyperplane arrangement. When $N=3$ we get a geodesic flow on the two-sphere minus three points. If, in addition we assume that the three masses are equal, then it was proved in [1] that the corresponding metric is hyperbolic: its Gaussian curvature is negative except at two points. Does the negative curvature property persist for $N=4$, that is, in the equal mass $1/r^2$ 4-body problem? Here we prove `no' by computing that the corresponding Riemannian metric in this $N=4$ case has positive sectional curvature at some two-planes. This `no' answer dashes hopes of naively extending hyperbolicity from $N=3$ to $N>3$.