Quote by Thurston on the Ricci flow I recall seeing a quote by William Thurston where he stated that the Geometrization conjecture was almost certain to be true and predicted that it would be proven by curvature flow methods. I don't remember the exact date, but it was from after Hamilton introduced the Ricci flow but well before Perelman's work. Unfortunately, most of the results for Geometrization and Ricci flow are from 2003 or after.  Does anyone know if the quote I'm referring to actually exists, and if so, where to find it?
There is a quote from Thurston lauding Perelman's work, which suggests that he thought the Ricci flow was a promising approach, but I thought there was one from before as well.

That the geometrization conjecture is true is not a surprise. That a
proof like Perelman's could be valid is not a surprise: it has a
certain rightness and inevitability, long dreamed of by many people
(including me). What is surprising, wonderful and amazing is that someone – Perelman – succeeded in rigorously analyzing and controlling this process, despite the many hurdles, challenges and potential pitfalls.

Thanks in advance.
 A: There is a video of Thurston's talk "A discussion on geometrization" from May 7, 2001. In the last part of this talk he speaks about possible approaches to proving geometrization.
Starting from 46:55 he spends 20 seconds mentioning heat type flows (probably Ricci flow) and says that it might or not work. But then he switches to describing a different approach which he thinks is more robust.
The part of the talk where he starts to speak about future predictions in the sense of proving geometrization starts at 39:32.
A: This 1994 paper by Thurston may or may not be the source you are thinking of, but it is a thoughtful essay that conveys the confidence Thurston had in his conjecture (albeit without referring to curvature flow):

The full geometrization conjecture is still a conjecture. It has been
proven for many cases, and is supported by a great deal of computer
evidence as well, but it has not been proven in generality. I am
convinced that the general proof will be discovered; I hope before too
many more years. At that point, proofs of special cases are likely to
become obsolete.

On proof and progress in mathematics
A: Most likely, this is just misremembering (or misattribution).

*

*In his math writing Thurston did not make any predictions regarding what approach to the Geometrization Conjecture (GC) will be successful. I did not hear him making such predictions in his math lectures (but, of course, I heard only few), and I would be surprised if he did. What Thurston might have said to somebody privately, I do not know.


*However, Thurston was well aware of the power of the Ricci Flow (RF) and in the early 1980s Thurston did write few pages with a sketch of the proof of the Orbifold Theorem which is the  geometrization theorem for orientable irreducible orbifolds with nonempty orbifold locus:



*

*W. P. Thurston, Three-manifolds with symmetry, preprint, Princeton University, 1982, 5 pages.
One of the steps of the proof was an application of Hamilton's 1982 result on manifolds of positive curvature (proven via RF, of course). A proof of the Orbifold Theorem was given later on in a work by Boileau, Leeb and Porti (broadly speaking, their proof follows Thurston's outline), shortly before appearance of Perelman's preprints. A bit different proof was later given in a work of Cooper, Hodgson and Kerckhoff.  Both proofs use Hamilton's theorem at some point, as Thurston suggested.


*In the mid 1990s Thurston was developing his own approach to the part of the GC known as the Hyperbolization Conjecture, based on analyzing laminations/foliations in 3-manifolds:


*

*Three-manifolds, Foliations and Circles, I. arXiv:math/9712268.



*In the last chapter of my 2001 book


*

*Kapovich, Michael, Hyperbolic manifolds and discrete groups, Progress in Mathematics (Boston, Mass.). 183. Boston, MA: Birkhäuser. xxv, 467 p. (2001). ZBL0958.57001.
I summarized  several known approaches to the GC. Two approaches were differential-geometric (RF and the Mike Anderson's approach), one was via min-volume (a combination of differential and Alexandrov geometry) and one was coarse geometric (to the "hyperbolic part" of the conjecture). At the time, it was quite possible that the GC will be proven in a piece-meal fashion with one proof of the hyperbolization conjecture, another for the Poincare Conjecture and yet another for Spherical Space Forms Conjecture.


*Regarding Ryan's comment: There was nothing even close to a consensus on the most promising approach to the GC. Hamilton and Yau believed it to be the RF, Mike Anderson had his own approach, Besson, Courtois and Gallot were making progress via the min-volume approach, some people in (or close to) Geometric Group Theory (like myself) believed it was via Cannon's Conjecture (still open) and Coarse Hyperbolization, etc. And there was no such  thing as "Thurston's program" for proving the GC. The closest thing to it was the plan for proving the Orbifold Theorem, but by the 1990s it was clear that this approach will not prove the full GC. In this list of successes one should also add the work of Gabai who in 1990s was proving directly some consequences of the Hyperbolization Conjecture, as well as the work of Tukia, Mess, Gabai, Casson and Jungreis, proving Seifert Conjecture.

