Abstract
In this paper we prove rigidity results on critical metrics for quadratic curvature functionals, involving the Ricci and the scalar curvature, on the space of Riemannian metrics with unit volume. It is well-known that Einstein metrics are always critical points. The purpose of this article is to show that, under some curvature conditions, a partial converse is true. In particular, for a class of quadratic curvature functionals, we prove that every critical metric with non-negative sectional curvature must be Einstein.
Similar content being viewed by others
References
Anderson, M.T.: Extrema of curvature functionals on the space of metrics on 3-manifolds. Calc. Var. Partial Differ. Equ. 5(3), 199–269 (1997)
Anderson, M.T.: Extrema of curvature functionals on the space of metrics on 3-manifolds. II. Calc. Var. Partial Differ. Equ. 12(1), 1–58 (2001)
Berger, M.: Quelques formules de variation pour une structure riemannienne. Ann. Sci. École Norm. Sup. 3(4), 285–294 (1970)
Besse, A.L.: Einstein Manifolds. Springer-Verlag, Berlin (2008)
Catino, G.: Critical metric of the \(L^{2}\)-norm of the scalar curvature. Proc. Am. Math. Soc. 142, 3981–3986 (2014)
Catino, G., Mastrolia, P., Monticelli, D.D.: A variational characterization of flat spaces in dimension three. arXiv preprint server (2014). http://arxiv.org
Gursky, M.J., Viaclovsky, J.A.: A new variational characterization of three-dimensional space forms. Invent. Math. 145(2), 251–278 (2001)
Gursky, M.J., Viaclovsky, J.A.: Rigidity and stability of Einstein metrics for quadratic curvature functionals. J. Reine Angew. Math. (2011). http://arxiv.org
Gursky, M.J., Viaclovsky, J.A.: Critical metrics on connected sums of Einstein four-manifolds (2013). arXiv preprint server http://arxiv.org
Hilbert, D.: Die Grundlagen der Physik. Ann. Sci. École Norm. Sup. 4, 461–472 (1915)
Hu, Z., Li, H.: A new variational characterization of \(n\)-dimensional space forms. Trans. Am. Math. Soc. 356(8), 3005–3023 (2003)
Hu, Z., Nishikawa, S., Simon, U.: Critical metrics of the Schouten functional. J. Geom. 98(1–2), 91–113 (2010)
Labbi, M.-L.: Variational properties of the Gauss-Bonnet curvatures. Calc. Var. Partial Differ. Equ. 32(2), 175–189 (2008)
Lamontagne, F.: Une remarque sur la norme \(L^2\) du tenseur de courbure. C. R. Acad. Sci. Paris Sér. I Math. 319(3), 237–240 (1994)
Lamontagne, F.: A critical metric for the \(L^2\)-norm of the curvature tensor on \(S^3\). Proc. Am. Math. Soc. 126(2), 589–593 (1998)
Smolentsev, N.K.: Spaces of Riemannian metrics. J. Math. Sci. 142(5), 2436–2519 (2007)
Tanno, S.: Deformations of Riemannian metrics on 3-dimensional manifolds. Tôhoku Math. J. 27(3), 437–444 (1975)
Viaclovsky, J.A.: Conformal geometry, contact geometry, and the calcu- lus of variations. Duke Math. J. 101(2), 283–316 (2000)
Viaclovsky, J.A.: The mass of the product of spheres (2013) arXiv preprint server http://arxiv.org
Acknowledgments
The author is members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Malchiodi.
Rights and permissions
About this article
Cite this article
Catino, G. Some rigidity results on critical metrics for quadratic functionals. Calc. Var. 54, 2921–2937 (2015). https://doi.org/10.1007/s00526-015-0889-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-015-0889-z