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 $\left( M,g\right) $ 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 $n\le 24$ by Khuri–Marques–Schoen (J Differ Geom 81(1):143–196, 2009), it has revealed to be generally false for $n\ge 25$ 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 $\frac{n-2}{4(n-1)} {{\mathrm{Scal}}}_g,\, {{\mathrm{Scal}}}_g$ being the Scalar curvature of $\left( M,g\right) $ . We show that a-priori $L^\infty $ –bounds fail for linear perturbations on all manifolds with $n\ge 4$ as well as a-priori gradient $L^2$ –bounds fail for non-locally conformally flat manifolds with $n\ge 6$ and for locally conformally flat manifolds with $n\ge 7$ . 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 $g$ .