A General Nonuniqueness Result for Yamabe-Type Problems for Conformally Variational Riemannian Invariants

Abstract

Abstract Given a conformally variational scalar Riemannian invariant I , we identify a sufficient condition for a compact Riemannian manifold to admit finite regular coverings with many nonhomothetic conformal rescalings with I constant. We also identify a sufficient condition for the universal cover to admit infinitely many nonhomothetic periodic conformal rescalings with I constant. Using these conditions, we improve known nonuniqueness results for the Q -curvatures of orders two, four, and six. We also prove nonuniqueness results for higher-order Q -curvatures and renormalized volume coefficients.