Sundman surfaces and Hill stability in the three-body problem

Using the famous Sundman inequality, we have constructed for the first time the surfaces for the general three-body problem that we suggest calling Sundman surfaces. These surfaces are a generalization of the widely known Hill surfaces in the restricted circular three-body problem. The Sundman surfaces are constructed in a rectangular coordinate system that uses the mutual distances between the bodies as the Cartesian rectangular coordinates. The singular points of the family of these surfaces have been determined. The possible and impossible regions of motion of the bodies have been constructed in the space of mutual distances. We have shown the existence of Hill stable motions and established sufficient criteria for Hill stability of motions. Some of the astronomical applications are considered.