Examples of non-isolated blow-up for perturbations of the scalar curvature equation on non locally conformally flat manifolds

TL;DR

This paper demonstrates that the typical isolated blow-up behavior of solutions to scalar curvature equations can fail under small smooth perturbations of the potential on certain manifolds, challenging existing compactness assumptions.

Contribution

It provides the first examples showing non-isolated blow-up points for perturbed scalar curvature equations on non-locally conformally flat manifolds.

Findings

Existence of non-isolated blow-up points under small perturbations

Breakdown of compactness in scalar curvature equations

Counterexamples in non-locally conformally flat settings

Abstract

Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.