2012/10/30 by Pierpaolo Esposito, Angela Pistoia, Esposito, Pierpaolo +3
Mathematics · #35B33 #35J61 #53C21 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · doi:10.48550/arxiv.1210.7979
openalex publication_date 2012/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 L2--bounds fail for non-locally 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.