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A geometric theory of zero area singularities in general relativity. (English) Zbl 1282.53059
A geometric theory of zero area singularities (ZAS) in Riemannian manifolds is developed generalizing the Schwarzschild metric to the case of negative mass. First, the notion of ZAS is introduced and an analysis of convergence for sequences of surfaces is presented. Two well-behaved classes of ZAS are defined: regular and harmonically regular. The mass of a ZAS is introduced as a numerical quantity that gives a lower bound on the Arnowitt-Deser-Misner (ADM) mass. The capacity of a ZAS is defined and it is shown that if it is positive then the mass is \(-\infty\). The spherically symmetric metrics with ZAS are studied as simple examples and their mass and capacity are calculated explicitly. The distinction between regular and harmonically regular ZAS is also emphasized.
The main result of the paper establishes a lower bound on the ADM mass for an asymptotically flat manifold of nonnegative scalar curvature in terms of masses of its singularities, assuming a certain conjecture in conformal geometry. It is proven that equality is attained in the inequality by the Schwarzschild metric of negative mass. The proof is given by using the Riemannian Penrose inequality in the case of positive mass and, in particular, certain types of incomplete metric are investigated. Several related open problems and conjectures are finally suggested. A detailed analysis of the asymptotically flat manifolds and conformal metrics in dimension 3 is presented in two appendices to the paper.

MSC:
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
83C99 General relativity
53C80 Applications of global differential geometry to the sciences
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