Is there a solution of the Yamabe problem using Ricci flow? 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.
 A: 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


*

*The
Ricci flow on surfaces (1988)

*The Ricci
flow on the 2-sphere (1991)

*The Yamabe flow on locally conformally flat manifolds with positive Ricci
curvature (1992)

*Global
existence and convergence of Yamabe flow (1994)
A: 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.
