Counter-examples to the generalized positive action conjecture
\textit{R. Schoen} and \textit{S. T. Yau} [Phys. Rev. Lett. 42, 547-548 (1979)] have shown that any complete asymptotically flat Riemannian 4-manifold (M,g) with zero scalar curvature has a mass, defined as \(\lim_{r\to \infty}\int_{S_ r}(g_{jk,k}-g_{kk,j})*dS^ j,\) nonnegative, and this mass vanishes iff g is flat. The generalized positive action conjecture of \textit{C. W. Hawking} and \textit{C. N. Pope} [Nucl. Phys. B 146, 381-392 (1978)] asserts that the above defined mass is non-negative for all locally asymptotically flat (M,g) with zero scalar curvature. The author of this paper constructs a counterexample for this Hawking- Pope conjecture and states the following theorem: ``The total space of any complex line bundle over \({\mathbb{C}}{\mathbb{P}}_ 1\) with Chern class \(c_ 1<-2\) carries a complete zero scalar curvature Kähler metric of negative mass. The generalized positive action conjecture is therefore false.''
asymptotically flat Riemannian 4-manifold
scientific article; zbMATH DE number 4078201