Uniqueness of critical metrics for a quadratic curvature functional

TL;DR

The paper addresses rigidity of complete critical metrics for the quadratic curvature functional $\mathfrak{S}^2$ in high dimensions. It derives the Euler–Lagrange equations, links negative or vanishing scalar curvature to a conformal quasi-Einstein framework, and proves that for $n \ge 10$ a complete metric with finite energy ($R \in L^q$ for suitable $q$) must be scalar-flat, hence a global minimum. The core strategy uses a conformal change $\widetilde{g} = |R|^{6/(n-4)} g$ to obtain a steady (quasi-)Einstein structure and leverages completeness and gradient estimates to rule out negative scalar curvature under the energy condition. Consequently, the results establish a sharp rigidity threshold in high dimensions, improving prior bounds and clarifying when finite energy forces scalar-flatness.

Abstract

In this paper we prove a new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functional $\mathfrak{S}^2 = \int R_g^{2} dV_g$: we show that critical metrics $(M^n, g)$ with finite energy are always scalar flat, i.e. global minima, provided $n\geq 10$.

Theorems & Definitions (11)

• Conjecture 1.1
• Theorem 1.2
• Lemma 2.1
• Corollary 2.2
• Proposition 2.3
• proof
• Corollary 2.4
• Proposition 3.1
• proof
• proof : Proof of Theorem \ref{['teo']}