Blow-up, homotopy and existence for periodic solutions of the planar three-body problem (Q1729659)

The author gives an expository proof of a result that he and \textit{R. Moeckel} [Nonlinearity 28, No. 6, 1919--1935 (2015; Zbl 1338.70018)] published in 2015 on the existence of reduced periodic orbits in the Newtonian planar three-body problem for equal or near-equal masses and angular momenta sufficiently small but nonzero, in every free homotopy class. This addresses the reduced version of the open question (posed in the paper): is every free homotopy class for the planar Newtonian three-body problem realized by a collision-free periodic solution? The author describes the breakthrough that led he and Moeckel to the result that uses the McGehee blow-up technique. The paper ends with a conjecture of the non-existence of periodic orbits when the angular momentum is zero. It presents four pieces of evidences to support this conjecture: 1) hyperbolic pants; 2) (not) hanging out at infinity; 3) Danya Rose's Bestiary; and 4) failure of the limits, i.e., the limits of the reduced periodic orbits that exist for non-zero angular momentum die in total collision as the angular momentum goes to zero. For the entire collection see [Zbl 1400.37004].