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Someone told me that it is possible to solve the Yamabe problem using Ricci flow. The proof I know of is the one originally proposed by Yamabe and then completed by Trudinger, Aubin and Schoen (in particular, the final step by Schoen made use of the positive mass theorem which had earlier been proved by Schoen and Yau).

If the Ricci flow proof exists, could someone point me to a reference?

Edit: When I say the 'Ricci flow' proof, I actually mean the Yamabe flow, since the two coincide on surfaces.

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2 Answers 2

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The references provided by Carlo Beenakker solve the problem only in special cases. The article "Global existence and convergence of Yamabe flow" by R. Ye assumes -- for example -- conformal flatness. The problem was solved completely within the last years. Important progress in dimension 3,4 and 5 was obtained by

  • Schwetlick and Struwe, Convergence of the Yamabe flow for "large'' energies. J. Reine Angew. Math. 562 (2003), 59–100.

and then later, the general statement was proven by Simon Brendle in

  • Brendle, Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170 (2007), no. 3, 541–576.

However, Brendle's proof only solves all cases, if one assumes the general version of the positive mass theorem. This general version was the subject of several preprints recently. Besides several articles by J. Lohkamp (see arXiv), there is also a preprint by Schoen and Yau https://arxiv.org/abs/1704.05490. To my knowledge no one of these preprints has been published so far.

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  • $\begingroup$ In the meantime, the article mentioned above was published as follows:Schoen, Richard(1-CA3); Yau, Shing-Tung(1-HRV) Positive scalar curvature and minimal hypersurface singularities.(English summary)Surveys in differential geometry 2019. Differential geometry, Calabi-Yau theory, and general relativity. Part 2, 441–480. Surv. Differ. Geom., 24 $\endgroup$ Feb 1 at 7:02
  • $\begingroup$ What is a bit unusual with this publication is: the article contains definitively great ideas about an important problem, thus it is important to communicate this to the community. However, it was published in a volume in honor of Yau's 70th birthday, although a solution of the positive mass theorem would deserve a much more prestigious journal. Unfortunately, there are no official explanations, and there are young mathematicians who feal unsure whether this result can be safely cited in view of this discrepancy. $\endgroup$ Feb 1 at 7:22
  • $\begingroup$ Rumors say that experts in geometric measure theory currently have mixed opinions on whether the measure theoretic arguments in this article are complete. For the differential geometers it would be interesting to learn more about this issue, to know whether the problem is settled or whether there are still open cases. Usually such questions are answered by accepting the preprint in a journal as J. Diff. Geom., Annals of Math., Inventiones etc., but it remains unclear why this preprint with great ideas and important consequences has gone another way. $\endgroup$ Feb 1 at 7:34
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I presume you are referring to the Yamabe flow approach to the Yamabe problem, which in 2 dimensions reduces to a Ricci flow. Relevant references include

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  • $\begingroup$ Yes, I meant the Yamabe flow and should have clarified this, thank you. $\endgroup$ Dec 8, 2019 at 23:34

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