10
$\begingroup$

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? I've tried Google and Arxiv searches and cannot find anything.

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

$\endgroup$
7
$\begingroup$

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 assume -- 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.

$\endgroup$
  • $\begingroup$ I thought Lohkamp had proved the positive mass theorem in all dimensions using a singularity excision argument, has this proof not been verified? I myself never even thought about doing any work on how to prove the general version of the positive mass theorem as I thought it had been dealt with by Lohkamp. $\endgroup$ – Tom Dec 9 '19 at 16:56
  • $\begingroup$ @Tom If you know how to prove it, you should write it up. $\endgroup$ – Ryan Unger Dec 14 '19 at 3:55
  • $\begingroup$ @Ryan Unger I'm not saying I know how to prove the positive mass theorem in arbitrary dimension. I thought about working on it, but ticked it off the list as having been done by someone else and case closed. $\endgroup$ – Tom Dec 14 '19 at 4:05
7
$\begingroup$

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

$\endgroup$
  • $\begingroup$ Yes, I meant the Yamabe flow and should have clarified this, thank you. $\endgroup$ – Tom Dec 8 '19 at 23:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.