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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.

MSC:

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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