Rigidity of Critical Metrics for Quadratic Curvature Functionals

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...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Catino, Giovanni, Mastrolia, Paolo, Dario Daniele Monticelli
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 06.10.2021
Subjects
Critical point
Curvature
Fields (mathematics)
Flat surfaces
Rigidity
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:2331-8422