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 (M, g) is compact. Established in the locally conformally flat case by Schoen [43, 44] and for n ≤ 24 by Khuri– Marques–Schoen [26], it has revealed to be generally false for n ≥ 25 as shown by Brendle [8] and Brendle–Marques [9]. A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential n−2 4(n−1) Scalg, Scalg being the Scalar curvature of (M, g). We show that a-priori L ∞–bounds fail for linear perturbations on all manifolds with n ≥ 4 as well as a-priori gradient L–bounds fail for nonlocally conformally flat manifolds with n ≥ 6 and for locally conformally flat manifolds with n ≥ 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.