Critical metrics of the $L^2$-norm of the scalar curvature
  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 critical metrics with nonnegative scalar curvature in dimension three and four. 
Giovanni Catino
math.DG math.AP