# What specifically is the gap in Aubin's argument about positive Ricci curvature that Paul Ehrlich alludes to?

In his paper [2], Paul Ehrlich write

In [1], Aubin stated a theorem which implied as a corollary that if a manifold $$M$$ admits a Riemannian metric with nonnegative Ricci curvature and all Ricci curvatures positive at some point, then $$M$$ admits a metric of everywhere positive Ricci curvature. It appears the proof in [1] is incomplete and the uniformity and correctness of Aubin's estimates even in the compact case are not clear.

Aubin paper is a bit technical so I want to know did Paul Ehrlich have a valid point, and if so, what is it, specifically?

What specifically is the gap in Aubin's argument about positive Ricci curvature that Paul Ehrlich alludes to?

Note 1: As far as I understand main part of the proof of wanted theorem is in pp. 397-399.

Note 2: Most of people that cited Aubin's paper (about positive Ricci curvature), also cited Paul Ehrlich's paper as well. Note also that most of papers that referred to Aubin's paper is because of results about scalar curvature and Yamabe problem.

[1]: Aubin, T., Métriques riemanniennes et courbure, J. Differ. Geom. 4, 383-424 (1970). ZBL0212.54102.

[2]: Ehrlich, Paul, Metric deformations of curvature. I: Local convex deformations, Geom. Dedicata 5, 1-23 (1976). ZBL0345.53024.

• @RyanBudney: Most of people cited Aubin paper, also cited Paul Ehrlich's paper as well. This makes me hesitate that perhaps others also have same problem of Paul Ehrlich. (I have no access to MathSciNet but I checked ZB). Jul 8 at 19:17
• @RyanBudney: FWIW, I think "is this result in a paper from the 70s correct" is pretty different from "is this result in a preprint that was posted to the arXiv yesterday correct," and doubly so if there is another published paper casting doubts on the result. Jul 8 at 19:34
• @RyanBudney: Maybe there should be a meta discussion. In my opinion, if the paper is more than a handful of years old, and you can either: point to a specific potential issue with the argument; or point to a public statement of doubt from someone else, then asking about the correctness of the paper is totally fine for MO. (These qualifiers eliminate 90% of the bad faith questions about new purported proofs of the Riemann hypothesis or whatever...) Jul 8 at 19:46
• I've put together a meta thread: meta.mathoverflow.net/questions/5390/… Jul 8 at 20:00
• during the classification of finite simple groups (CFSG) effort, in the 70s-80s, there were a number of not too correct papers in this areas. And CFSG was declared complete on basis on a never published hundreds pages long preprint. Jul 22 at 8:39