## G. Catino - P. Mastrolia - Dario Daniele Monticelli

# Rigidity of Critical Metrics for Quadratic Curvature Functionals

created by catino on 17 Oct 2023

[

BibTeX]

*preprint*

**Inserted:** 17 oct 2023

**Last Updated:** 17 oct 2023

**Year:** 2021

**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.