On the stability of $$L^p$$ L p -norms of Riemannian curvature at rank one symmetric spaces

Abstract

We study stability and local minimizing property of $$L^p$$ -norms of Riemannian curvature tensor denoted by $$\mathcal {R}_p$$ by variational methods. We compute the Hessian of $$\mathcal {R}_p$$ at compact rank 1 symmetric spaces and prove that they are stable for $$\mathcal {R}_p$$ for certain values of $$p\ge 2$$ . A similar result also holds for compact quotients of rank 1 symmetric spaces of non-compact type. Consequently, we obtain stability of $$L^{\frac{n}{2}}$$ -norm of Weyl curvature at these metrics using results from Gursky and Viaclovsky (J Reine Angew Math 400:37–91, 2015).