Results on the existence of the Yamabe minimizer of M^m times R^n

We let (M^m, g) be a closed smooth Riemannian manifold (m >1) with positive scalar curvature S_g, and prove that the Yamabe constant of (M \times R^n,g+g_E) is achieved by a metric in the conformal class of (g+g_E), where g_E is the Euclidean metric. We also show that the Yamabe quotient of (M \times R^n,g+g_E) is improved by Steiner symmetrization with respect to M. It follows from this last assertion that the dependence on R^n of the Yamabe minimizer of (M \times R^n,g+g_E) is radial.