[EDITED to make the question more suitable for MO. See meta.mathoverflow.net for discussion about re-opening.]
"I am not certain that the proof is totally nailed down. … there are very few experts in the area of Ricci flow, and I have not yet met anyone who claims to have a complete understanding of the last, most difficult part of Perelman's proof … As far as I'm aware, no one has taken some of the techniques Perelman introduced toward the end of his paper and successfully used them to solve any other significant problem. This suggests to me that other mathematicians don't yet have full command of this work and its methodologies either."
To what extent are the doubts that Yau expressed well-founded?
Here is a fuller quotation of the relevant passage from Yau's book The Shape of a Life (pages 258–260).
One problem I'm not actively working on is the Poincaré conjecture, as I'm happy to put the controversy surrounding it behind me. But I can't keep my mind from turning to that problem, upon occasion, and I still have some lingering doubts that—if expressed out loud—are likely to get me in trouble. Although it may be heresy for me to say this, I am not certain that the proof is totally nailed down. I am convinced, as I've said many times before, that Perelman did brilliant work regarding the formation and structure of singularities in three-dimensional spaces—work that was indeed worthy of the Fields Medal he was awarded (but chose not to accept). Perelman built upon a foundation painstakingly laid down by Hamilton and carried us further along the path laid out by Poincaré than we've ever ventured before. About this I have no doubts, and for that, Perelman deserves tremendous credit. Yet, I still wonder how far his work involving Ricci flow "technology" has taken us. And I also can't keep from wondering whether another approach—making use of some of the minimal surface techniques I developed many years ago with Bill Meeks, Rich Schoen, and Leon Simon—might lend some clarity to the situation.
In 2003, Perelman told Dana Mackenzie, a reporter for Science magazine, that it would be "premature" to make a public announcement regarding a proof of the geometrization and Poincaré conjectures until other experts in the field weighed in on the matter. Confirmation of this proof resided largely with outside "experts," given that Perelman receded almost completely from the mathematics scene, which is a great loss to the field. The thing is, there are very few experts in the area of Ricci flow, and I have not yet met anyone who claims to have a complete understanding of the last, most difficult part of Perelman's proof.
In 2006 or thereabouts, a visiting mathematician who was knowledgeable about this area stopped by my Harvard office to reproach me for raising questions about Perelman's work. Yet he admitted, when I asked him, that he did not entirely grasp the latter part of Perelman's argument. That's no knock on him, as that admission puts him in a rather sizable group. In fact, I don't know whether anyone else, including Hamilton, has fully gotten it, and I'd put myself in that category as well. As far as I'm aware, no one has taken some of the techniques Perelman introduced toward the end of his paper and successfully used them to solve any other significant problem. This suggests to me that other mathematicians don't yet have full command of this work and its methodologies either.
Hamilton, who's now in his seventies, has told me that it is still his dream to prove the Poincaré conjecture. That does not mean that he thinks Perelman did anything wrong. Hamilton, a truly independent spirit, is not one to follow in someone else's footsteps, nor would he be inclined to "connect the dots" of another's argument. He just may want to do it his own way and complete his life's work of the past three and a half decades.
Nevertheless, that still leaves me with the sense that this situation is not unequivocally resolved, perhaps leaving theorems of incredibly broad sweep hanging in the balance. Expressing my doubts on this subject, I know from experience, is a politically fraught proposition. But for the sake of my own questions—and for mathematics as a whole—I'd still like to be more certain of where we stand. If that makes me a pariah, so be it. In the end I care more about mathematics—the path I chose to follow more than a half century ago—than I do about what others think of me.