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-g0 is positive. In this paper, we show the sign condition of R-g0 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) is an element of M with R-g0 (x(0)) > 0.