Critical metrics of the L^2-norm of the scalar curvature

Details Critical metrics of the L^2-norm of the scalar curvature Critical metrics of the L^2-norm of the scalar curvature

2012-04-12

arxiv.org/abs/1204.2744v1

Mathematics

Differential Geometry

Scientific paper

In this paper we investigate complete critical metrics of the $L^{2}$-norm of the scalar curvature. We prove that any complete critical metric with positive scalar curvature has constant scalar curvature and we characterize complete critical metrics with nonnegative scalar curvature in dimension three and four. Moreover, by means of the estimates developed in the proof, we prove a rigidity result for static vacuum solutions in any dimension, which in turn implies that any complete $(0,n+1)$-Einstein manifold is Ricci flat.

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