Existence of conformal metrics with constant \(Q\)-curvature. (English) Zbl 1186.53050

On a four-dimensional compact Riemannian manifold \((M, g)\), the conformally invariant Paneitz operator \(P_g\) in a certain sense provides a generalization of the Laplace-Beltrami operator on a surface. Under a conformal change of metric \(\tilde{g}= e^{2w}g\), the associated \(Q\)-curvature \(Q_g\) is related to \(P_g\) through the well known formula \(P_gw+ 2Q_g= 2Q_{\tilde g}e^{4w}\), which implies that the integral \(k_p\) of \(Q_g\) is also a conformal invariant.
There have been many useful results on the geometric properties of \(P_g\) and \(Q_g\). Taking account of one of M. J. Gursky [Commun. Math. Phys. 207, No. 1, 131–143 (1999; Zbl 0988.58013)], the authors generalize one of the main results in [S.- Y. A. Chang and P. C.- P. Yang, Ann. Math. (2) 142, No. 1, 171–212 (1995; Zbl 0842.58011)], concerning the analogue in dimension four to the uniformization theorem for Riemannian surfaces.
Namely, it is proved that if \(\ker P_g= \{\mathrm{constants}\}\) and \(k_p \neq 8k{\pi}^2\) for \(k= 1, 2,\dots\), then \((M, g)\) admits a conformal metric with constant \(Q\)-curvature. As these assumptions are conformally invariant and generic, the result applies to a rather large class of 4-manifolds, including some of negative curvature, but does not cover the case of locally conformally flat manifolds with positive and even Euler characteristic.


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35B33 Critical exponents in context of PDEs
35J35 Variational methods for higher-order elliptic equations
53C20 Global Riemannian geometry, including pinching
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58J70 Invariance and symmetry properties for PDEs on manifolds
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