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

TL;DR

This paper proves the existence of Yamabe minimizers on product manifolds involving a closed manifold with positive scalar curvature and Euclidean space, showing symmetrization improves the Yamabe quotient and the minimizer's radial dependence.

Contribution

It establishes the existence of Yamabe minimizers on (M^m × R^n) and demonstrates that Steiner symmetrization enhances the Yamabe quotient, revealing the radial nature of the minimizer.

Findings

Yamabe constant is achieved by a conformal metric on M×R^n.

Steiner symmetrization improves the Yamabe quotient.

Yamabe minimizer depends radially on R^n.

Abstract

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.