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

Abstract

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 $$x_0 \in M$$ with $$R_{g_0}(x_0) > 0.$$