Rigidity and Non-rigidity Results on the Sphere

It is a simple consequence of the maximum principle that a superharmonic function u on Rn(i. e. ∆u ≤ 0) which is 1 near infinity is identically 1 on Rn (throughout this paper, n ≥ 3). Geometrically this means that one can not conformally deform the Euclidean metric in a bounded region without decreasing the scalar curvature somewhere. In fact there is a much stronger result: one can not have any compact deformation of the Euclidean metric without decreasing the scalar curvature somewhere, i. e. , if g is a metric on Rn which has nonnegative scalar curvature and is the Euclidean metric near infinity, then g is the Euclidean metric on Rn. This is a simple version of the positive mass theorem ([9, 12]). Another implication of the positive mass theorem is the following rigidity theorem for the unit ball in Rn.