Green functions and conformal geometry. (Q1609756)

A canonical metric, which is unique in its conformal class is constructed here with the Green function of the Yamabe operator (conformal Laplacian) and used for investigating the moduli space of locally conformally flat structures. This construction depends on the positive mass theorem of Schoen-Yau and the resulting metric is different from those obtained earlier by other methods. The behavior of the canonical metric under surgery-type degeneration is analyzed and the main result says that the limit of the canonical metrics yields the canonical metric of the limit space. Consequently, the \(L^2\)-metric on the moduli space of scalar positive locally conformally flat structures is not complete and the example of \(S^1 \times S^2\) as underlying manifold is studied.