Counter-examples to the generalized positive action conjecture

Abstract

We give examples of complete locally asymptotically flat Riemannian 4-manifolds with zero scalar curvature and negative mass. The generalized positive action conjecture of Hawking and Pope [5] is therefore false.The positive action theorem, proved by Schoen and Yau [9], states that any complete asymptotically flat Riemannian 4-manifold (M, g) with scalar curvature R = 0 satisfies lim $(g jk . k -g kktj )*dS j ^Q r-*ac S r with equality iff g is flat; here the integrand is to be computed in an asymptotic coordinate system in which the metric is of the form ~2 ana S r denotes the Euclidean sphere of radius r. For simplicity we will call the left-hand side of the above inequality the mass of (M, g), since this expression is the analog of the ADM mass of an asymptotically flat 3-manifold. The generalized positive action conjecture of Hawking and Pope [5] asserts that the mass is also non-negative for all locally asymptotically flat Riemannian 4-manifolds with scalar curvature R = 0, and that, moreover, the mass vanishes iff the manifold is Ricci flat with self-dual Weyl curvature.Unfortunately, as we will see, this plausible-sounding extension does not hold water. We will produce an infinite number of complete locally asymptotically flat Riemannian 4-manifolds with R = 0 for which the mass is negative. These metrics are Kahler and live on the total spaces of complex line-bundles over S 2 -CP : for which the first Chern class satisfies c 1 < -2. They have isometry group (7(2), and, by virtue of being Kahler with R -0, have anti-self-dual Weyl curvature; cf. [2,7,8].

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