The existence of generalized isothermal coordinates for higher-dimensional Riemannian manifolds

Abstract

We shall show that, for any given point p on a Riemannian manifold (M, g ), there is a pointwise conformai metric g = <i>g in which the ^-geodesic sphere centered at p with radius r has constant mean curvature 1/r for all sufficiently small r . Furthermore, the exponential map of g at p is a measure preserving map in a small ball around p .