Abstract
Point-like masses on a given initial domain of the complex plane are interacting globally under gravity and as a consequence the initial domain is mapped into a final domain which encodes all effects of localized sources . The mapping between the two domains is a truncated beta function which so far has not been fully investigated. In this paper we propose to know more about the final domain and so we start from the remark that when the masses are put on the real axis of the upper half plane, the mapping in question is the Schwarz-Christoffel mapping (SCM), a popular and useful tool in computational geometry. The linking of planar gravity with the SCM will allow us to use all what we know from this mapping to get answer for certain static solutions in planar gravity. Among the problems we will study we mention: masses distributed on the real axis of the full complex plane, masses distributed on a circle, the dust string either closed or infinite open. Strings with tensions are however not linked to SCM as these mappings do not involve the concept of tension.
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