On Variational Properties of Quadratic Curvature Functionals

TL;DR

This paper analyzes the stability of quadratic curvature functionals on closed manifolds, focusing on their behavior at metrics with constant sectional curvature, contributing to understanding geometric variational problems.

Contribution

It introduces a detailed study of the variational properties and stability of quadratic curvature functionals at constant curvature metrics.

Findings

Identifies conditions for stability of quadratic curvature functionals.

Provides criteria for when these functionals are minimized or maximized.

Enhances understanding of geometric variational problems on manifolds.

Abstract

In this paper, we investigate a class of quadratic Riemannian curvature functionals on closed smooth manifold $M$ of dimension $n\ge 3$ on the space of Riemannian metrics on $M$ with unit volume. We study the stability of these functionals at the metric with constant sectional curvature as its critical point.