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Critical point equation on four-dimensional compact manifolds. (English) Zbl 1307.58006

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.
Denote 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 \[ {\mathcal L}_g^*(f)=\overset\circ{Ric}(*) \] where \(\overset\circ{Ric}\) is the traceless Ricci tensor and \(f\) a smooth function. A CPE 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 \((*)\).
In 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.

MSC:

58E11 Critical metrics
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58D17 Manifolds of metrics (especially Riemannian)
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