Sign-changing blow-up for the Yamabe equation at the lowest energy level

Abstract

We investigate the blow-up behavior of sequences of sign-changing solutions for the Yamabe equation on a Riemannian manifold (M,g) of positive Yamabe type. For each dimension n≥11, we describe the value of the minimal energy threshold at which blow-up occurs. In dimensions 11≤n≤24, where the set of positive solutions is known to be compact, we show that the set of sign-changing solutions is not compact and that blow-up already occurs at the lowest possible energy level. We prove this result by constructing a smooth, non-locally conformally flat metric on space forms Sn/Γ, Γ≠{1}, whose Yamabe equation admits a family of sign-changing blowing-up solutions. As a counterpart of this result, we also prove a sharp compactness result for sign-changing solutions at the lowest energy level, in small dimensions or under strong geometric assumptions.