Open Access
October, 2016 Stable solutions of the Yamabe equation on non-compact manifolds
Jimmy PETEAN, Juan Miguel RUIZ
J. Math. Soc. Japan 68(4): 1473-1486 (October, 2016). DOI: 10.2969/jmsj/06841473


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 [2], 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.


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Jimmy PETEAN. Juan Miguel RUIZ. "Stable solutions of the Yamabe equation on non-compact manifolds." J. Math. Soc. Japan 68 (4) 1473 - 1486, October, 2016.


Published: October, 2016
First available in Project Euclid: 24 October 2016

zbMATH: 1355.53032
MathSciNet: MR3564440
Digital Object Identifier: 10.2969/jmsj/06841473

Primary: 53C21
Secondary: 53C20

Keywords: Non-compact manifolds , Yamabe equation

Rights: Copyright © 2016 Mathematical Society of Japan

Vol.68 • No. 4 • October, 2016
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