Is it possible to show the existence of harmonic coordinates (e.g., on uniform-sized balls) on certain classes of non-compact Riemannian manifolds? For example, one may expect that such harmonic coordinates exist on a manifold that has doubling volume measure and supports Poincare inequality?



harmonic coordinates on non-compact manifolds
The Main Lemma 2.2 in "Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds" (<http://link.springer.com/article/10.1007/BF01233434>) says essentially that there is a uniform lower bound for the harmonic radius in term of Ricci curvature bound and lower bound of injectivity radius.


The paper "E. HEBEY & M. HERZLICH, Harmonic coordinates, harmonic radius and convergence of Riemannian manifolds" (<http://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/1997(4)/569-605.pdf>) is an exposition of the above paper of Anderson, which is, in my viewpoint, easier to read.


