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

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

2009-11-30

arxiv.org/abs/0911.5724v1

Mathematics

Differential Geometry

Scientific paper

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.

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