Absence of free-time minimizers for N-body problems and a splitting theorem

For the $N$-body problem in dimension $d\ge 3$, we give an alternative proof of a theorem of da Luz and Maderna on the nonexistence of free-time minimizers defined on the entire real line, and extend it from the Newtonian potential to a larger class of singular subharmonic pairwise interaction potentials. The proof is based on the following generalization of the Cheeger-Gromoll splitting theorem: let $(M,g)$ be a connected Riemannian manifold and let $U$ be a smooth positive function such that ${\mathrm{R}\mathrm{i}\mathrm{c}}_g\ge \frac{\mathrm{\Delta }U}{2U}g$, and the metrics $Ug$ and $U^{-1}g$ are both complete; if the metric $Ug$ admits a line, then the function $U$ is constant and $(M,g)$ splits off that line. We apply this splitting theorem to the Jacobi-Maupertuis metric associated with the $N$-body problem; a quantitative version of Marchal’s theorem enables us to reduce to the case of smooth potentials.