The effect of linear perturbations on the Yamabe problem

TL;DR

This paper investigates how linear perturbations affect the compactness properties of Yamabe metrics, revealing that certain bounds fail in various dimensions and geometric contexts, thus providing new insights into the stability of solutions.

Contribution

It demonstrates the failure of a-priori bounds under linear perturbations for Yamabe problems across different dimensions and geometries, extending understanding of the problem's stability.

Findings

A-priori L^0-bounds fail for all manifolds with na04.

Gradient L^2-bounds fail for non-locally conformally flat manifolds with na06.

Results are optimal in several geometric and dimensional settings.

Abstract

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\leq 24 by Khuri-Marques-Schoen [26], it has revealed to be generally false for n\geq 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) Scal_g, Scal_g being the Scalar curvature of (M,g). We show that a-priori L^\infty-bounds fail for linear perturbations on all manifolds with n\geq 4 as well as a-priori gradient L^2--bounds fail for non-locally conformally flat manifolds with n\geq 6 and for locally conformally flat manifolds with n\geq 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.