Gradient estimates for potentials of invertible gradient-mappings on the sphere

McCann showed that, if the potential of a gradient-mapping, on a compact riemannian manifold, is c-convex, the length of its gradient cannot exceed the diameter of the manifold. We improve this bound in two different manners on the constant curvature spheres, under assumptions on the relative density of the image-measure of the riemannian volume. One proof, with the standard metric, relies on the Brenier-McCann optimal measure-transport property; the other, purely pde, ignores it. This work is thought of as a preliminary step toward a second derivatives estimate which would yield smooth optimal measure-transport (open).