A general nonuniqueness result for Yamabe-type problems for conformally variational Riemannian invariants

Author:

João Henrique Andrade, Jeffrey S. Case, Paolo Piccione, Juncheng Wei

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Analysis of PDEs (math.AP)

journal:

date:

2023-10-23 16:00:00

Abstract

Given a conformally variational scalar Riemannian invariant $I$, we identify a sufficient condition for a closed 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 geometrically distinct periodic conformal rescalings with $I$ constant. Using these conditions, we improve known nonuniqueness results for the $Q$-curvatures of orders two, four, and six, and establish nonuniqueness results for higher-order $Q$-curvatures and renormalized volume coefficients.

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