Stable solutions of the Yamabe equation on non-compact manifolds

We consider the Yamabe equation on a complete non compact Riemannian manifold and study the condition of stability of solutions. If (M-m,g) is a closed manifold of constant positive scalar curvature, which we normalize to be m(m - 1), we consider the Riemannian product with the n-dimensional Euclidean space: (M-m x R-n, g + gE). And we study, as in [2], the solution of the Yamabe equation which depends only on the Euclidean factor. We show that there exists a constant lambda(m, n) such that this solution is stable if and only if lambda(1) >= lambda(m, n), where lambda(1) is the first positive eigenvalue of -Delta(g). We compute lambda(m, n) numerically for small values of m, n showing in these cases that the Euclidean minimizer is stable in the case M = S-m with the metric of constant curvature. This implies that the same is true for any closed manifold with a Yamabe metric.