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.


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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[1] Anderson, Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds, Geom. Funct. Anal. 9 pp 855– (1999) · Zbl 0976.53046
[2] Atiyah, Self-duality in four-dimensional Riemannian geometry, Proc. R. Soc. Lond. 362 pp 425– (1978) · Zbl 0389.53011
[3] Besse, Einstein Manifolds (2008)
[4] Cao, On locally conformally flat gradient steady Ricci solitons, Trans. Am. Math. Soc. 364-5 pp 2377– (2012) · Zbl 1245.53038
[5] J. Chang S. Hwang G. Yun Total scalar curvature and harmonic curvature 2011
[6] X. Chen Y. Wang On four-dimensional anti-self-dual gradient Ricci solitons 2011
[7] Cheng, Eigenvalues and nodal sets, Comment. Math. Helv. 51 pp 43– (1976) · Zbl 0334.35022
[8] Corvino, Scalar curvature deformations and a gluing construction for the Einstein constraint equations, Commun. Math. Phys. 214 pp 137– (2000) · Zbl 1031.53064
[9] Fischer, Manifolds of Riemannian metrics with prescribed scalar curvature, Bull. Am. Math. Soc. 80 pp 479– (1974) · Zbl 0288.53040
[10] Gilbarg, Elliptic Partial Differential Equations of 2nd Order (1983) · Zbl 0562.35001
[11] Hwang, Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature, Manuscr. Math. 103 pp 135– (2000) · Zbl 0972.58009
[12] Hwang, The critical point equation on a three-dimensional compact manifold, Proc. Am. Math. Soc. 131 pp 3221– (2003) · Zbl 1046.53015
[13] Hwang, Rigidity of the critical point equation, Math. Nachr. 283 pp 846– (2010) · Zbl 1198.53065
[14] Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan 34 pp 665– (1982) · Zbl 0486.53034
[15] Lafontaine, Sur la géométrie d’une généralisation de l’équation différentielle d’Obata, J. Math. Pures Appl. 62 pp 63– (1983)
[16] Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 pp 333– (1962) · Zbl 0115.39302
[17] Qing, A note on static spaces and related problems, J. Geom. Phys. 74 pp 18– (2013) · Zbl 1287.83016
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