Green functions and conformal geometry, I + II
Lutz Habermann and Jürgen Jost
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Submission date: 28. Feb. 1997
published in: Journal of differential geometry, 53 (1999) 3, p. 405-443
with the following different title: Green functions and conformal geometry
We use the Green function of the conformal 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 conformal 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 conformal maps. We analyze the behavior of our metric if the scalar curvature tends to 0. The example of S¹ × S² as underlying manifold is studied in detail.
We demonstrate that the canonical metrics associated to scalar positive locally conformally flat structures introduced in the first paper in this series 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.