The effect of linear perturbations on the Yamabe problem

In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold is compact. Established in the locally conformally flat case by Schoen (Lecture Notes in Mathematics, vol. 1365, pp. 120-154. Springer, Berlin 1989, Surveys Pure Application and Mathematics, 52 Longman Science, Technology, pp. 311-320. Harlow 1991) and for by Khuri-Marques-Schoen (J Differ Geom 81(1):143-196, 2009), it has revealed to be generally false for as shown by Brendle (J Am Math Soc 21(4):951-979, 2008) and Brendle-Marques (J Differ Geom 81(2):225-250, 2009). A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential being the Scalar curvature of . We show that a-priori -bounds fail for linear perturbations on all manifolds with as well as a-priori gradient -bounds fail for non-locally conformally flat manifolds with and for locally conformally flat manifolds with . In several situations, the results are optimal. Our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of g.