On the stability of the L ^p-norm of the Riemannian curvature tensor

Abstract

We consider the Riemannian functional $\mathcal {R}_{p}(g)={\int }_{M}|R(g)|^{p}dv_{g}$ defined on the space of Riemannian metrics with unit volume on a closed smooth manifold M where R(g) and dv _g denote the corresponding Riemannian curvature tensor and volume form and p ∈ (0, ∞). First we prove that the Riemannian metrics with non-zero constant sectional curvature are strictly stable for $\mathcal {R}_{p}$ for certain values of p. Then we conclude that they are strict local minimizers for $\mathcal {R}_{p}$ for those values of p. Finally generalizing this result we prove that product of space forms of same type and dimension are strict local minimizer for $\mathcal {R}_{p}$ for certain values of p.