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Stable solutions of the Yamabe equation on non-compact manifolds. (English) Zbl 1355.53032

Summary: 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: \((M^m\times\mathbb{R}^n,g+g_E)\). And we study, as in [K. Akutagawa et al., Commun. Anal. Geom. 15, No. 5, 947–969 (2007; Zbl 1147.53032)], the solution of the Yamabe equation which depends only on the Euclidean factor. We show that there exists a constant \(\lambda(m,n)\) such that this solution is stable if and only if \(\lambda_1\geq\lambda(m,n)\), where \(\lambda_1\) is the first positive eigenvalue of \(-\Delta_g\). We compute \(\lambda(m,n)\) 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.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C20 Global Riemannian geometry, including pinching

Citations:

Zbl 1147.53032
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