Critical point equation on four-dimensional compact manifolds (Q2929422)

The integral of the scalar curvature, \({\mathcal S}(g)=\int_MR_gdM_g\), as a functional on the space \(\mathcal M\) of Riemannian metrics of unit volume on a compact manifold \(M\) is very well known, dating back to Hilbert, and the critical points are the Einstein metrics. Here the authors consider the same functional on the space \(\mathcal C\) of Riemannian metrics of constant scalar curvature.NEWLINENEWLINEDenote by \({\mathcal L}_g\) the linearization of the scalar curvature operator and by \({\mathcal L}_g^*\) its formal \(L^2\)-adjoint. The Euler-Lagrange equation for \({\mathcal S}(g)\) restricted to \(\mathcal C\) may be written as NEWLINE\[NEWLINE{\mathcal L}_g^*(f)=\overset\circ{Ric}\leqno(*)NEWLINE\]NEWLINE where \(\overset\circ{Ric}\) is the traceless Ricci tensor and \(f\) a smooth function. A CPE \textit{metric} is a 3-tuple \((M^n,g,f)\) where \((M^n,g)\) is a compact oriented Riemannian manifold of dimension \(\geq 3\) with constant scalar curvature and \(f\) is a non-constant smooth potential satisfying equation \((*)\).NEWLINENEWLINEIn general when one considers a critical point problem for a curvature functional on a given set of metrics, if one restricts the functional to a smaller set of metrics, one would expect a weaker critical point condition. However it has been conjectured by several authors that a CPE metric is Einstein. The main result of the present paper is that this conjecture is true for 4-dimensional half conformally flat manifolds. Recall that a Riemannian metric on a 4-dimensional manifold is half conformally flat if it is either self-dual or anti-self-dual.