Uniqueness of Positive Solutions of the Conformal Scalar Curvature Equation and Applications to Conformal Transformations

We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We also study uniqueness of complete positive solutions and radial solutions.