ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS OF MOBIUS EQUATIONS

A generalization of the Schwarzian derivative to conformal mappings of Riemannian manifolds has naturally introduced the corresponding overdetermined differential equation which we call the Mobius equation. We are interested in study of the existence and uniqueness of the solution of the Mobius equation. Among other things, we show that, for a compact manifold, if Ricci curvature is nonpositive, for a complete noncompact manifold, if the scalar curvature is a positive constant, then the differential equation has only constant solutions. We also study the nonhomogeneous equation in an n-dimensional Euclidean space.