The Yamabe problem for Q-curvature

Details The Yamabe problem for Q-curvature The Yamabe problem for Q-curvature

2011-10-20

arxiv.org/abs/1110.4615v2

Mathematics

Analysis of PDEs

This paper has been withdrawn by the author because "On the $k$th order Yamabe problem" has made it obsolete

Scientific paper

In this paper we demonstrate that under general conditions there exists a metric in the conformal class of an arbitrary metric on a smooth, closed Riemannian manifold of dimension greater than four such that the $Q$-curvature of the metric is a constant. Existence of solutions is obtained through the combination of variational methods, second order Sobolev inequalities, and the $W^{2,2}$ blow-up theory developed by Hebey and Robert. Positivity of the solutions is obtained from a novel argument proven here for the first time that is rooted in the conformal covariance property of the Paneitz-Branson operator and the positive semidefiniteness of the second derivative of a $C^2$ function at a local minimum.

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