Rigidity of Critical Metrics for Quadratic Curvature Functionals

TL;DR

This paper establishes new rigidity results for complete critical metrics of quadratic curvature functionals, showing that under various conditions, such metrics must be flat or scalar flat, with proofs involving conformal vector fields, Anderson's ideas, and new estimates.

Contribution

The paper provides novel rigidity theorems for critical metrics of quadratic curvature functionals, extending known results to non-compact cases and higher dimensions with new analytical techniques.

Findings

Flat surfaces are the only critical points of -squared scalar curvature functional.

Flat 3D manifolds are the only critical points of the quadratic curvature functional for t > -1/3.

3D scalar flat manifolds are the only critical points of -squared scalar curvature with finite energy.

Abstract

In this paper we prove new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functionals $\mathfrak{F}^{2}_t = \int |\operatorname{Ric}_g|^{2} dV_g + t \int R^{2}_g dV_g$, $t\in\mathbb{R}$, and $\mathfrak{S}^2 = \int R_g^{2} dV_g$. We show that (i) flat surfaces are the only critical points of $\mathfrak{S}^2$, (ii) flat three-dimensional manifolds are the only critical points of $\mathfrak{F}^{2}_t$ for every $t>-\frac{1}{3}$, (iii) three-dimensional scalar flat manifolds are the only critical points of $\mathfrak{S}^2$ with finite energy and (iv) $n$-dimensional, $n>4$, scalar flat manifolds are the only critical points of $\mathfrak{S}^2$ with finite energy and scalar curvature bounded below. In case (i), our proof relies on rigidity results for conformal vector fields and an ODE argument; in case (ii) we draw upon some ideas of M. T. Anderson concerning regularity, convergence and rigidity of critical metrics; in cases (iii) and (iv) the proofs are self-contained and depend on new pointwise and integral estimates.