On the problem of convexity for the restricted three-body problem around the heavy primary

Abstract

In this note, we prove that below the first critical energy level, a proper combination of the Ligon-Schaaf and Levi-Civita regularization mappings provides a convex symplectic embedding of the double cover of the energy surfaces of the planar rotating Kepler problem into ${\mathbb R}^{4}$ endowed with its standard symplectic structure. This convex embedding extends to the bounded component of the planar circular restricted three-body problem around the heavy body outside a small neighborhood of the collisions. This opens up new approaches to attack the Birkhoff conjecture about the existence of a global surface of section in the restricted planar circular three-body problem using holomorphic curve techniques.