A quick search on Amazon provides at least three titles that are introductory texts to the topic for graduate students.
(1) B. Chow, P. Lu, L. Ni: Hamilton's Ricci Flow, Graduate Studies in Mathematics 77, AMS 2006;
(2) B. Chow, D. Knopf: The Ricci Flow: An Introduction, Mathematical Surveys and Monographs 110, AMS 2004;
(3) B. Chow and others: The Ricci Flow: Techniques and Applications: Geometric Aspects, Mathematical Surveys and Monographs 135, AMS 2007.
First, let me make some preliminary remarks. We sometimes like to think that "being proved" is a black-and-white property, but in fact there are shades of gray. At one extreme are things like the infinitude of the primes, whose proof every mathematician understands. But then there are results that are widely accepted but no proof has appeared. Vladimir Voevodsky has pointed out that "a technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail," and that this practice can lead to false statements being erroneously accepted as proved; recognizing this point led Voevodsky to spend much of the later part of his career on computer verification of formal proofs. Although the mathematical literature is generally very reliable, it is far from perfect; this point has been addressed in more detail in another MO question about the extent of wrong papers in research mathematics. So in
vast majority of cases, "being proved" isn't about being 100% confident
that there is no error; it's about whether the proof has been sufficiently
scrutinized that the chances of a serious mistake are negligible.
Returning to Yau,
if you look carefully at what he is saying, you will see that, technically, he does not say that he thinks Perelman's proof is wrong, or that it has a serious gap, or even that there are parts of the proof that nobody understands. He says only that he is not certain that the proof is totally nailed down, and that he has not met anyone who understands the most difficult part of the proof. He also points out that if a powerful new idea is properly digested by the mathematical community then it usually leads to the solution of new problems, and that if this has not happened with the most difficult part of Perelman's proof then it probably means that this part of the proof deserves more study.
In principle, calling for the mathematical community to devote more time to studying an important and difficult proof in order to "nail it down" and acquire a "complete understanding" and a "full command of this work" is unobjectionable. In the past, I have heard colleagues say that the original work of various Fields Medalists—Hironaka and Freedman come to mind—was very difficult to understand and that there was a need for the community to study and assimilate those groundbreaking ideas more thoroughly. In both the cases of Hironaka and Freedman, the community has indeed put in effort to study their work, and rich dividends have resulted, so this type of activity is definitely worth encouraging. Note that this doesn't mean that the original proofs were wrong or had serious gaps; it just means that the proofs moved closer to the infinitely-many-primes ideal of universal understanding, and the chance of an unnoticed significant gap or error was driven down even closer to zero.
Unfortunately, Yau chose to phrase his remarks in a "politically fraught" manner that he knew would "get him in trouble."
He says things in a way that (probably intentionally)
gives many readers the impression
that he is casting doubt on the correctness and completeness of Perelman's proof (even though, as I said, technically he doesn't explicitly say that the proof is wrong or incomplete). The book appeared in 2019 but the most recent conversation he cites was from 2006.
He makes no mention of recent research in the area which
does in fact apply Perelman's ideas to solve new problems.
It should therefore not be surprising that the consensus of the mathematical community is that Yau's remarks do not pose any serious challenge to the conclusion that Perelman's proofs—especially of the Poincaré Conjecture, which involves fewer technicalities than the Geometrization Conjecture—are correct. There were at least three separate efforts which came to this conclusion. Kleiner and Lott's detailed notes say, regarding Perelman's original papers [51] and [52]:
Regarding the proofs, the papers [51, 52] contain some incorrect statements and incomplete arguments, which we have attempted to point
out to the reader. (Some of the mistakes
in [51] were corrected in [52].) We did not find any serious problems, meaning problems
that cannot be corrected using the methods introduced by Perelman.
Similarly, Morgan and Tian wrote:
In this book we present a complete and detailed proof of the
Poincaré Conjecture. … The arguments we give here are a detailed version of those that appear in Perelman’s three preprints.
There is also the account of Huai-Dong Cao and Xi-Ping Zhu, which Yau himself refereed.
On top of those three detailed accounts of Perelman's proof, there have been more recent developments. Terry Tao mentions the recent survey by Richard Bamler. Moishe Kohan mentions Kleiner and Lott's Geometrization of Three-Dimensional Orbifolds via Ricci Flow and Bamler and Kleiner's proof of the Generalized Smale Conjecture. So contrary to the impression you might form from what Yau said, the community is indeed continuing to milk Perelman's ideas and apply them to solving new problems. If there are specific technical points which Yau thinks are obscure, I am sure that other researchers would be happy to address them if Yau were to spell them out explicitly. Until then, there is no credible reason to doubt the fundamental correctness of Perelman's arguments.
Best Answer
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
and then later, the general statement was proven by Simon Brendle in
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.