Scalar curvature rigidity for asymptotically locally hyperbolic manifolds. (English) Zbl 0946.53021

A Riemannian manifold is called asymptotically locally hyperbolic, if it has an end where the metric is asymptotic to the quotient of hyperbolic space by a finite isometric group action. Rigidity results for asymptotically hyperbolic spin manifolds with lower bound on the scalar curvature are proved using spinorial methods related to Witten’s proof of the positive mass theorem. The existence of an asymptotic imaginary Killing spinor leads to an imaginary Killing spinor, which allows to use a classification result by H. Baum on manifolds with imaginary Killing spinors. The expression for the mass is given and it is shown that the mass vanishes for conformally compact Einstein manifolds with conformal boundary a spherical space form.


53C27 Spin and Spin\({}^c\) geometry
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
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