Stable solutions of the Yamabe equation on non-compact manifolds

TL;DR

This paper investigates the stability of solutions to the Yamabe equation on non-compact manifolds formed by products of closed manifolds with Euclidean space, identifying a critical eigenvalue threshold for stability.

Contribution

It introduces a stability criterion based on the first eigenvalue and computes a key constant numerically for small dimensions, extending stability results to general Yamabe metrics.

Findings

Existence of a critical eigenvalue mbda(m,n) for stability.

Numerical computation of mbda(m,n) for small m,n.

Euclidean minimizers are stable on spheres with constant curvature.

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: $(M^m \times \mathbf{R}^n, g+ g_E)$. And study the solution of the Yamabe equation which depends only on the Euclidean factor. We show that there exists a constant $\lambda (m,n)$ such that the 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.