Given a closed Riemannian manifold, two conformal metrics with the same constant scalar curvature may still be
geometrically distinct, that is, not related by a conformal diffeomorphism followed by a rescaling. We prove that products of
spheres with hyperbolic manifolds admit countably many pairwise nonhomothetic complete periodic metrics with constant
scalar curvature. The key principle is that the properness of the conformal group action, guaranteed by the Ferrand--Obata
theorem, combined with a volume shrinkage argument along a tower of finite coverings, provides an obstruction to conformal
homothety. We then extend this framework to conformally variational invariants, obtaining nonuniqueness theorems for curvature
prescription problems, including $Q$-curvatures, $\sigma_k$-curvatures, and renormalized volume coefficients.
This is a joint work with J. S. Case, P. Piccione, and J. Wei.