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

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

► Let (Mm,g)(Mm,g) be a compact, smooth, Riemannian manifold with positive scalar curvature. ► Let (N,h)=(MmxRn,g+gE), with n,m>1n,m>1, and gEgE the Euclidean metric. ► Steiner symmetrization with respect to MM improves the Yamabe functional of (N,h)(N,h). ► The Yamabe minimizer of (N,h)(N,h) exists, and is positive and smooth. ► The dependence on RnRn of the Yamabe minimizer of (N,h)(N,h) is radial.