Counter-examples to the generalized positive action conjecture
type: Item
instance of: scholarly article
title: Counter-examples to the generalized positive action conjecture
zbMATH Open document ID: 0659.53050
DOI: 10.1007/BF01221110
author: Claude Lebrun
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: \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.''
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 53C99
zbMATH DE Number: 4078201
zbMATH Keywords: asymptotically flat Riemannian 4-manifold
zbMATH Keywords: counterexample
zbMATH Keywords: Hawking-Pope conjecture
zbMATH Keywords: complex line bundle
zbMATH Keywords: negative mass
zbMATH Keywords: generalized positive action conjecture
reviewed by: Daniel H. Gottlieb
MaRDI profile type: Publication
Wikidata QID: Q123012039
cites work: Self-dual solutions to Euclidean gravity
cites work: Q4846449
cites work: Polygons and gravitons
full work available at URL: https://doi.org/10.1007/bf01221110
OpenAlex ID: W2071845785
description: scientific article; zbMATH DE number 4078201