## 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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### References:

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