On compact conformally flat 4-manifolds
Let \(M\) be a compact connected manifold and \(C\) a conformal class of metrics on \(M\). The resolution of the Yamabe problem asserts that, for any \((M,C)\), there exists a metric \(g\) contained in \(C\) such that the volume of \((M,g)\) is equal to 1 and the scalar curvature of \(g\) is constant and equal to the Yamabe invariant \(\mu(M,C)\) of \((M,C)\). Let \(M\) be an oriented 4-manifold with positive Euler number and suppose that \((M,C)\) is conformally flat. Denote by \(C_ 0\) the conformal class containing the standard metric on \(S^ n\). Then the result of the present paper is stated as follows: (1) If \(\mu(M,C) > -\mu(S^ n,C_ 0)\), then \((M,C)\) is conformal to \((S^ n,C_ 0)\). (2) If \(\mu(M,C) = - \mu(S^ n,C_ 0)\), then \(C\) contains a negative constant curvature metric. A similar result for non-orientable 4-manifolds is also proved in the paper.
scientific article; zbMATH DE number 495487
scientific article