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

TL;DR

This paper studies the blow-up behavior of sign-changing solutions to the Yamabe equation on certain manifolds, identifying energy thresholds for blow-up and constructing examples demonstrating non-compactness in specific dimensions.

Contribution

It establishes the minimal energy level at which blow-up occurs for sign-changing solutions in dimensions 11 to 24 and constructs explicit metrics with blowing-up solutions, advancing understanding of solution behavior.

Findings

Blow-up occurs at the lowest energy level for sign-changing solutions in dimensions 11 to 24.

Constructed non-locally conformally flat metrics with sign-changing blowing-up solutions.

Proved compactness results for sign-changing solutions under certain geometric conditions.

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 $n\ge11$, we describe the value of the minimal energy threshold at which blow-up occurs. In dimensions $11 \le n \le 24$, 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 $\mathbb{S}^n/\Gamma$, $\Gamma \neq \{1\}$, 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.