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A General Nonuniqueness Result for Yamabe-Type Problems for Conformally Variational Riemannian Invariants
João H Andrade1, Jeffrey S Case2, Paolo Piccione1,3,4
1Present Address: Institute of Mathematics and Statistics, University of São Paulo, São Paulo, SP 05508-090 Brazil.
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.
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