Nonuniqueness of constant scalar curvature metrics in a conformal class.

摘要

The Yamabe Problem asks when the conformal class of a compact, Riemannian manifold (M, g) contains a metric g¯ of constant scalar curvature. Its resolution established that it always does, and that the sign of this constant is a conformal invariant.;We consider metrics whose conformal class includes a metric of constant positive scalar curvature. We show that given a smooth manifold ( M, g) of dimension n ≥ 9, there exists a metric g˜ which is arbitrarily close to g in the C1,alpha topology and whose conformal class contains an arbitrary number of distinct metrics with constant scalar curvature equal to 1. If we assume, in addition, that (M, g) is locally conformally flat, we may take g˜ to be close to g in the Cs topology for any s n2.;These results generalize, in dimensions n ≥ 9, earlier results of Ambrosetti, Ambrosetti and Malchiodi, Berti and Malchiodi, and Pollack. Our proof constructs parameterized perturbations of an explicit approximate solution. The conformal class containing the constant scalar curvature metrics is obtained in this manner, and so has a well-understood geometry.