Title: Fitting hyperbolic pants to a 3-body problem.

Abstract: We investigate the three-body problem with an attractive $1/r^2$ potential. (The Newtonian gravitational potential is $1/r$.) Modulo symmetries, the dynamics of the bounded zero-angular momentum solutions is equivalent to a geodesic flow on the thrice-punctured sphere, or ``pair of pants''. This metric generating these geodesics is complete and infinite area. Our main result is that if the three masses are equal then the metric is hyperbolic (negative curvature) except at two points (the Lagrange points). Corollaries are a nearly complete symbolic-dynamical understanding of the zero-angular momentum bounded dynamics for equal masses. In particular we get:
(1) uniqueness, mod symmetries, of the $1/r^2$ figure eight,
(2)existence and uniqueness, mod symmetries, of a periodic solution for almost every free homotopy class of the pants,
(3) a complete symbolic dynamics encoding bounded non-collision orbits,
(4) denseness of collision orbits.

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