The Obata-Vétois argument and its applications

We extend Vétois' Obata-type argument and use it to identify a closed interval I_n, n ≥ 3, containing zero such that if a ∈ I_n and (Mⁿ, g) is a compact conformally Einstein manifold with nonnegative scalar curvature and Q₄ + ασ₂ constant, then it is Einstein. We also relax the scalar curvature assumption to the nonnegativity of the Yamabe constant under a more restrictive assumption on α. Our results allow us to compute many Yamabe-type constants and prove sharp Sobolev inequalities on compact Einstein manifolds with nonnegative scalar curvature. In particular, we show that compact locally symmetric Einstein four-manifolds with nonnegative scalar curvature extremize the functional determinant of the conformal Laplacian, partially answering a question of Branson and Ørsted.