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I'm trying to read the paper of Jost and Karcher on the existence of harmonic coordinates on a ball whose size only depend on the injectivity radius and a two sided bound on the curvature.

Unfortunately, my german skills are quite low and make the reading really slow. Does there exist another place to find this proof (in English or French) ?


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You ever tried to ask him personally? He is a very nice person! – Orbicular Dec 7 '10 at 16:39
Although it's probably too late... I learned a proof from M. Anderson's paper, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math., 1990. (Main Lemma 2.2) – Chih-Wei Chen Sep 26 '11 at 1:33
up vote 4 down vote accepted

Perhaps Jost's account of it from his lectures on harmonic mapppings between Riemannian manifolds contains a detailed description.

There's also some related results due to Hebey and Herzlich

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First, I can confirm that the construction and estimates of almost linear and harmonic co-ordinates appear on the monograph of Jost cited by Willie. I also believe that Anderson and Cheeger prove the existence of harmonic co-ordinates on uniform sized balls using a blow-up argument in their 1992 JDG paper. – Deane Yang Dec 7 '10 at 18:02
Thanks, I found Hebey and Herzlich in the library, it seems nicely written. According to the authors, their proof is the same as the one by Anderson and Cheeger. I'll try to understand these. About Jost monograph, I'll try to have a look on it but it's not in the library of my lab, so it will take longer. Thanks again. – Thomas Richard Dec 7 '10 at 21:55

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