Results on the existence of the Yamabe minimizer of Mm×Rn

We let ( M m , g ) be a closed smooth Riemannian manifold with positive scalar curvature S g , and prove that the Yamabe constant of ( M × R n , g + g E ) ( n , m ≥ 2 ) is achieved by a metric in the conformal class of ( g + g E ) , where g E is the Euclidean metric. We do this by showing that the Yamabe functional of ( M × R n , g + g E ) is improved under Steiner symmetrization with respect to M , and so, the dependence on R n of the Yamabe minimizer of ( M × R n , g + g E ) is radial.