Greenfunctions and conformal geometry

Abstract

We use the Green function of the Yamabe operator (conformai Laplacian) to construct a canonical metric on each locally conformally flat manifold different from the standard sphere that supports a Riemannian metric of positive scalar curvature. In dimension 3, the assumption of local conformai flatness is not needed. The construction depends on the positive mass theorem of Schoen-Yau. The resulting metric is different from those obtained earlier by other methods. In particular, it is smooth and distance nondecreasing under conformai maps. We analyze the behavior of our metric if the scalar curvature tends to 0. We demonstrate that the canonical metrics converge under surgery-type degenerations to the corresponding metric on the limit space. As a consequence, the L2—metric on the moduli space of scalar positive locally conformally flat structures is not complete. The example of S1 × S2 as underlying manifold is studied in detail. © 1999 Applied Probability Trust.