On the stability of 𝐿𝑝 -norms of Riemannian curvature at rank one symmetric spaces

Abstract

We study stability and local minimizing property of 𝐿𝑝-norms of Riemannian curvature tensor denoted by 𝑝 by variational methods. We compute the Hessian of 𝑝 at compact rank 1 symmetric spaces and prove that they are stable for 𝑝 for certain values of 𝑝≥2. A similar result also holds for compact quotients of rank 1 symmetric spaces of non-compact type. Consequently, we obtain stability of 𝐿𝑛2-norm of Weyl curvature at these metrics using results from Gursky and Viaclovsky (J Reine Angew Math 400:37–91, 2015).