Conformal Rigidity and Non-rigidity of the Scalar Curvature on Riemannian Manifolds

For a compact smooth manifold ( M , g 0 ) with a boundary, we study the conformal rigidity and non-rigidity of the scalar curvature in the conformal class. It is known that the sign of the first eigenvalue for a linearized operator of the scalar curvature by a conformal change determines the rigidity/non-rigidity of the scalar curvature by conformal changes when the scalar curvature R g 0 is positive. In this paper, we show the sign condition of R g 0 is not necessary, and a reversed rigidity of the scalar curvature in the conformal class does not hold if there exists a point x 0 ∈ M with R g 0 ( x 0 ) > 0 .