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The positive mass theorem for ALE manifolds. (English) Zbl 0890.53065

Chruściel, Piotr T. (ed.), Mathematics of gravitation. Part I: Lorentzian geometry and Einstein equations. Proceedings of the workshop on mathematical aspects of theories of gravitation, Warsaw, Poland, February 29–March 30, 1996. Warsaw: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 41(1), 133-142 (1997).
The author considers the question of extending the positive mass argument of E. Witten [Commun. Math. Phys. 80, 381-402 (1981; Zbl 1051.83532)] to an asymptotically locally Euclidean (ALE) manifold. This involves the imposition of an additional condition; more specifically, the author shows that the ‘generalized positive action conjecture’ holds when the signature of the manifold has the correct value. Contents include: preliminaries (giving an overview of spin geometry); asymptotically locally Euclidean manifolds; ALE from curvature decay; spinors and the Lichnerowicz formula; the positive mass theorem; and specialization to four dimensions.
For the entire collection see [Zbl 0880.00054].

MSC:

53Z05 Applications of differential geometry to physics
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
83C30 Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory

Citations:

Zbl 1051.83532
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