Higher-order Obata–Vetois rigidity for constant Q-curvature

Establish whether the Obata–Vetois uniqueness theorem extends to Q-curvatures of order higher than six for conformal metrics on closed Einstein manifolds with positive scalar curvature, beyond the sixth-order case proved here.

Background

The paper studies conformal metrics with constant sixth-order Q-curvature on closed connected positive Einstein manifolds and proves that every such metric is Einstein, thereby resolving the sixth-order instance of a broader uniqueness problem. The relevant rigidity principle originates with Obata for scalar curvature and was extended by Vetois to fourth-order Q-curvature.

The authors place their result within the broader question of extending this rigidity theorem to higher-order Q-curvatures. Since the manuscript resolves only the sixth-order case, the extension to still higher orders remains the explicitly identified unresolved problem.