Absence of free-time minimizers for N-body problems and a splitting theorem
Abstract
For the N-body problem in dimension d ≥ 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 Ricg ≥ ΔU/2Ug, and the metrics Ug and U−1g 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.
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BibTeX
@misc{assouline2026absence,
title = {{Absence of free-time minimizers for N-body problems and a splitting theorem}},
author = {Rotem Assouline and Marco Mazzucchelli},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2610.00959},
eprint = {2610.00959},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05777457v1},
}
RIS
TY - UNPB
TI - Absence of free-time minimizers for N-body problems and a splitting theorem
AU - Rotem Assouline
AU - Marco Mazzucchelli
PY - 2026
DA - 2026/09/24
PB - HAL
DO - 10.48550/arxiv.2610.00959
UR - https://hal.science/hal-05777457v1
AB - For the N-body problem in dimension d ≥ 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 Ric_g ≥ ΔU/2Ug, and the metrics Ug and U^−1g 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.
ER -