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 $(M,{g^0})$, there is a pointwise conformal metric $g = \Phi {g^0}$ in which the $g$-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 $p$.