The Obata–Vétois argument and its applications

Abstract We extend Vétois’ Obata-type argument and use it to identify a closed interval I n I₍, n ≥ 3 n 3, containing zero such that if a ∈ I n a I₍ and (M n, g) (M^n, g) is a compact conformally Einstein manifold with nonnegative scalar curvature and Q 4 + a ⁢ σ 2 Q₄+a₂ 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 𝑎. 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.