On the Ricci and Einstein equations on the pseudo-euclidean and hyperbolic spaces

Abstract

We consider tensors T = f g on the pseudo-euclidean space R n and on the hyperbolic space H n , where n ⩾ 3 , g is the standard metric and f is a differentiable function. For such tensors, we consider, in both spaces, the problems of existence of a Riemannian metric g ¯ , conformal to g, such that Ric g ¯ = T , and the existence of such a metric which satisfies Ric g ¯ − K ¯ g ¯ / 2 = T , where K ¯ is the scalar curvature of g ¯ . We find the restrictions on the Ricci candidate for solvability and we construct the solutions g ¯ when they exist. We show that these metrics are unique up to homothety, we characterize those globally defined and we determine the singularities for those which are not globally defined. None of the non-homothetic metrics g ¯ , defined on R n or H n , are complete. As a consequence of these results, we get positive solutions for the equation Δ g u − n ( n − 2 ) 4 λ u ( n + 2 ) ( n − 2 ) = 0 , where g is the pseudo-euclidean metric.