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

Abstract

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold $M$ given by ${\mathcal{R}}_{\frac{n}{2}}(g):={\int }_M|R(g){|}^{\frac{n}{2}}d{v}_g$ where $R(g)$, $d{v}_g$ 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.

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