Stability of Quadratic curvature Functionals at product Einstein manifolds

TL;DR

This paper investigates the stability of quadratic curvature functionals at product Einstein manifolds, revealing instability conditions related to the spectral properties of hyperbolic components.

Contribution

It provides new stability criteria for quadratic curvature functionals at product Einstein manifolds, especially involving eigenvalues of hyperbolic factors.

Findings

Product of spherical and hyperbolic manifolds can be unstable for certain quadratic functionals.

Instability depends on the first Laplacian eigenvalue of the hyperbolic component.

The study advances understanding of curvature functional stability in geometric analysis.

Abstract

In this paper, we study Riemannian functionals defined by $L^2$-norms of Ricci curvature, scalar curvature, Weyl curvature, and Riemannian curvature. We try to understand stability of their critical points that are products of Einstein metrics. In particular, we prove that the product of a spherical space form and a compact hyperbolic manifold is unstable for some quadratic functionals if the first eigenvalue of the Laplacian of the hyperbolic manifold is sufficiently small.