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

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 R4 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 col-lisions. 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.