Stable solutions of the Yamabe equation on non-compact manifolds

Abstract

Abstract. We consider the Yamabe equation on a complete non-compact Riemannian manifold and study the condition of stability of solutions. If (M m , g) is a closed manifold of constant positive scalar curvature, which we normalize to be m(m − 1), we consider the Riemannian product with the n-dimensional Euclidean space: (M m ×R n , g +g E ). And study, as in [2], the solution of the Yamabe equation which depends only on the Euclidean factor. We show that there exists a constant λ(m, n) such that the solution is stable if and only if λ 1 ≥ λ(m, n), where λ 1 is the first positive eigenvalue of −∆ g . We compute λ(m, n) numerically for small values of m, n showing in these cases that the Euclidean minimizer is stable in the case M = S m with the metric of constant curvature. This implies that the same is true for any closed manifold with a Yamabe metric.