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Correctness of Thierry 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 is P.did Paul Ehrlich claim/objectionhave a valid point, and if so, what is it, specifically?

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

Note 1: As far as I understand main part of Aubin'sthe proof acceptable?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.

Correctness of Thierry Aubin's argument about positive Ricci curvature

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 is P. Ehrlich claim/objection about correctness or incompleteness of Aubin's proof acceptable?

[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.

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.

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C.F.G
C.F.G
  • 4.2k
  • 6
  • 31
  • 65

Correctness of Thierry Aubin's argument about positive Ricci curvature

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 is P. Ehrlich claim/objection about correctness or incompleteness of Aubin's proof acceptable?

[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.