I would disagree with Ben McKay's comment a little bit, in that whenever Ricci flow produces something like a "best metric," you must be on a very special manifold and so from a certain perspective the "best metric" already trivially exists. But this point is more metaphysical than mathematical. It might be more useful to point out that all of Hamilton and Perelman's work on singularity formation is very important and useful and reveals deep structures, but does not actually produce best metrics, natural forms, equilibrium states etc., even in the form of the Thurston conjecture itself

@DeaneYang Bishop announced his work in the 1963 Notices of the AMS

There is a small error in your post, the given solution is not on R^2, it is only defined on the unit disk or a subset thereof. It also doesn't make sense to take t to infinity (what is the underlying space?). Usually steady solitons are defined for all time, it is not the case here since the metric is not complete. Any complete steady soliton has nonnegative scalar curvature. (theorem of Bing-Long Chen)

Thanks, that looks like a great paper. I see there's also an extension of the usual Hopf-Rinow theorem. Although that does answer the version of the theorem I put in my question, I am still curious about the more general version.

My impression is that this is done on page 628 of Coifman and Weiss "Extensions of Hardy spaces and their use in analysis," but I'm not familiar with any details

Clearly enough my concerns aren't shared by others, I've deleted my previous comments

@WillSawin Perelman's papers have their own errors, which is of course unremarkable (this is why I mentioned #5). So I suppose what you say doesn't apply, although I've never understood such analyses

@IgorBelegradek I couldn't personally say, since I have met too few people who have even made the attempt to read all of it.

I'm only acknowledging it as already mentioned on the page linked to in the question. It's not a source.

To say the least, there are many Ricci flow experts who only understand some certain small parts of Perelman's second paper. Anyway, depending on the researcher's field there are two possible types of research-level questions, one about specific details and one about the existence of expert consensus. To me, and only speaking for myself, the latter question is rather complicated in the present case, and I find most responses to it rather superficial. That's just to say that I would benefit from a thoughtful answer to some version of this question, and that I've hoped for many years to read it!

@MoisheKohan As someone familiar with many parts of Hamilton and Perelman's work but not an up-to-date expert in the field, it seems clear that there is a research community with somewhere on the order of five members, maybe fewer, who use and develop the most complicated parts of Perelman's work. As such, Yau's use of "no one" seems unjustified but his point seems easily adaptable. To me, it seems that many (even most) parts of Perelman's work are widely understood but that some parts are still obscure to the vast majority, which seems notable for such major work from 20 years ago