On The Stability of The L^p Norm of The Curvature Tensor

TL;DR

This paper studies the stability of the L^p norm of the curvature tensor on closed manifolds, showing that constant curvature metrics are strictly stable and local minima of this functional.

Contribution

It proves the strict stability and local minimality of constant curvature metrics for the L^p curvature norm functional on closed manifolds.

Findings

Constant curvature metrics are critical points of the functional.

Such metrics are strictly stable under the L^p curvature norm.

These metrics are local minima in their neighborhood.

Abstract

We investigate stability and local minimizing properties of the Riemannian functional defined by the L^p norm of the curvature tensor on the space of Riemannian metrics on a closed manifold. Riemannian metrics with constant curvature and products of such metrics are critical points of this functional. We prove that these points are strictly stable for this functional and if (M; g) is a manifold of this type, g has a neighborhood U such that g is the strict minima on it.