Some Unstable Critical Metrics for $L^{\frac{n}{2}}$-norm of the Curvature Tensor

TL;DR

This paper investigates the stability of certain geometric functionals related to the curvature tensor on closed manifolds, revealing that some symmetric spaces are unstable critical points for these functionals.

Contribution

It identifies specific locally symmetric spaces that serve as unstable critical points for the $L^{n/2}$-norm of the curvature tensor functional.

Findings

Locally symmetric spaces can be unstable critical points.

The functional studied is the $L^{n/2}$-norm of the curvature tensor.

Unstable critical points are explicitly characterized.

Abstract

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold $M$ given by $\mathcal{R}_{\frac{n}{2}}(g):= \int_M |R(g)|^{\frac{n}{2}}dv_g$ where $R(g)$, $dv_g$ denote the Riemannian curvature and volume form corresponding to $g$. We show that there are locally symmetric spaces which are unstable critical points for this functional.