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

TL;DR

This paper investigates the stability of L^p-norms of the Riemannian curvature tensor at rank one symmetric spaces, demonstrating stability for certain p values and extending results to non-compact quotients, with implications for Weyl curvature.

Contribution

It computes the Hessian of the L^p-norm of curvature at rank one symmetric spaces and proves their stability for specific p, extending to non-compact quotients.

Findings

Stability of L^p-norms at rank one symmetric spaces for p > 2

Hessian calculations confirm local minimality of curvature norms

Stability results extend to non-compact quotients of symmetric spaces

Abstract

We study stability and local minimizing properties 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 > 2. A similar result also holds for compact quotients of rank 1 symmetric spaces of non-compact type. Consequently, we obtain stability of L^{n\2}- norm of Weyl curvature at these metrics.