We extend Vétois’ Obata-type argument and use it to identify a closed interval <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>I</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> Iₙ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> n≥ 3 , containing zero such that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>a</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mi>I</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:math> a∈ Iₙ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>M</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (Mⁿ,g) is a compact conformally Einstein manifold with nonnegative scalar curvature and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>Q</m:mi> <m:mn>4</m:mn> </m:msub> <m:mo>+</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo></m:mo> <m:msub> <m:mi>σ</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:mrow> </m:math> Q₄+aσ₂ 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.
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Jeffrey S. Case (2024) studied this question.