Stable solutions of the Yamabe equation on non-compact manifolds

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 × ℝⁿ, g + g_E). And we 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 this solution is stable if and only if λ₁ ≥ λ (m,n), where λ₁ is the first positive eigenvalue of −Δ_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.