The gradient of certain harmonic functions on manifolds of almost nonnegative Ricci curvature

In this paper we prove that when the Ricci curvature of a Riemannian manifoldMn is almost nonnegative, and a ballBL(p)⊂Mn is close in Gromov-Hausdorff distance to a Euclidean ball, then the gradient of the harmonic functionb defined in [ChCo1] does not vanish. In particular, these functions can serve as harmonic coordinates on balls sufficiently close to an Euclidean ball. The proof, is based on a monotonicity theorem that generalizes monotonicity of the frequency for harmonic functions onRn.