Some unstable critical metrics for the $L^{\frac{n}{2}}$-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 R n 2 (g) := M |R(g)| n 2 dvg where R(g), dvg 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.

Stability of Quadratic curvature Functionals at product Einstein manifolds

“…Recently, stability of rank 1 symmetric spaces as critical points of R and W n 2 has been established in [13]. The behaviour of irreducible symmetric spaces of higher rank as critical metrics of R is not completely understood, but some unstable critical points of the functional R n 2 have been pointed out in [4]. The above theorem implies that if the first eigenvalue of the laplacian of a hyperbolic manifold H n is sufficiently small and n > 5 then its product with S n is unstable for R.…”

On the stability of L^p-norms of Curvature Tensor at Rank one symmetrics spaces

“…To study H p on the space of conformal variations of g we use Proposition (1.1) in [5]. We observe that if f is an eigenfunction corresponding to the first positive eigenvalue of the Laplacian of HP m or OP 2 then H 2 (f g, f g) is negative.…”

“…To study H p on the space of conformal variations of g we recall the following result from [5]. Proposition 1.…”