sentences:  The basic regularity theorem concerning the metric in harmonic coordinates is that if the components of the metric are in the Hölder space "C""k",α when expressed in "some" coordinate system, then they are in that same Hölder space when expressed in harmonic coordinates.Harmonic mapping to generate harmonic coordinates in regions with boundary is one of the original well known methods for grid generation in the field of computational fluid dynamics.Here the goal is to find a harmonic map of a given region (in Euclidean space or in a Riemannian manifold) to a convex region (very often a rectangle or a box in the case of grid generation in Euclidean space) with the additional requirement that the boundary map should be a homeomorphism (see the works of S. S. Sritharan in the reference list below).