Stable solutions of the Yamabe equation on non-compact manifolds

2015/02/04 by Petean, Jimmy, Ruiz, Juan Miguel
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1502.01092

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 study 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 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.

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