Speaker
Description
Critical metrics for the Hilbert-Einstein functional in dimension three are very rigid since they are necessarily of constant sectional curvature. Substituting the scalar curvature by a quadratic curvature invariant, one is leaded to the consideration of the functionals
$$
\mathcal{S}: g\mapsto\mathcal{S}(g)=\int_M\tau_g dvol_g\,,
\qquad
\mathcal{F}_t:g\mapsto\mathcal{F}_t(g)=\int\{\|\rho_g\|^2+t\tau_g^2\}dvol_g,
$$
where $\rho$ and $\tau$ denote the Ricci tensor and the scalar curvature of the metric $g$.
These functionals have been extensively studied in mathematics and physics (see, for example [1,2,3]).
The purpose of this lecture is to present some new results on the classification of critical metrics both in the homogeneous and curvature-homogeneous settings. A specific feature of the Lorentzian situation is the existence of non-Einstein metrics which are critical for all quadratic curvature functionals.
References:
- M. Berger, Quelques formules de variation pour une structure riemannienne,
\emph{Ann. Sci. École Norm. Sup.} \textbf{3} (1970), no. 4, 285--294. - E. A. Bergshoeff, O. Hohm, and P. K. Townsend, Massive gravity in three dimensions, \emph{Phys. Rev. Lett.} \textbf{102} (20) (2009), 201301, 4 pp.
- M. J. Gursky and J. A. Viaclovsky, A new variational characterization of three-dimensional space forms, \emph{Invent. Math.} \textbf{145} (2001), 251--278.
