Stability of quadratic curvature functionals at product of Einstein manifolds

Abstract

We consider Riemannian functionals defined by $$L^2$$ -norms of Ricci curvature, scalar curvature, Weyl curvature, and Riemannian curvature. Rigidity, stability, and local minimizing properties of Einstein metrics as critical metrics of these quadratic functionals have been studied in 8. In this paper, we study the same for products of Einstein metrics with Einstein constants of possibly opposite signs. In particular, we show that the Riemannian product of closed Einstein manifolds $$(M_0,g_0)$$ and $$(M_1,g_1)$$ with respective Einstein constants $$\lambda (>0)$$ and $$-\lambda $$ , is unstable for certain quadratic functionals if the first eigenvalue of the Laplacian of $$(M_1,g_1)$$ is sufficiently small. We also prove the stability of $$L^{\frac{n}{2}}$$ -norm of Weyl curvature at compact quotients of $$S^n\times {\mathbb {H}}^m$$ .