Influence of Yau's solution to the Calabi Conjecture on the field of PDEs I remember reading a long time ago(I can't recall where, unfortunately) that Yau's solution of the Calabi-Yau conjecture introduced new techniques that were very important for the field of partial differential equations. 
My question is essentially two-fold: first of all, what exactly were the breakthroughs that he made, and secondly, why do they have a wide applicability to other PDEs? My background is in geometry, so I'm well-aware of the importance of the confirmation of the Calabi conjecture, but I'm interested in the analytic side here. I know that he applied the continuity method, and used some very hard a priori bounds to show closedness of the set of solutions, the openness having been proved by Calabi, but not much more.
 A: Probably not quite the answer to your question,  nevertheless some historical remarks: the barrier function employed by Yau for deriving the estimates on the Laplacian was already used by Progorelov in his resolution of the Minkowski problem.
In that approach,  one reduces the Laplacian estimate to the zero order estimates, that is, to  $\sup |\phi|$. This was the missing piece in deriving the estimates. 
The case of finding K\"ahler-Einstein metrics, when $c_1<0$, was in that respect easier as the first order estimates are pretty easy to derive. This was independently done by Aubin as well. But rather than the continuity method, Aubin uses a more difficult variational approach.
Yau's method for finding the zeroth order estimate was pretty involoved, which was later simplified by Kazdan and Bourguignon. Notice that if one wanted to solve the equation on a domain with boundary rather than a closed manifold, the $C^0$ estimate again comes for free.
The third order estimates are also not straightforward  to derive. The quantity, $S$, that was considered by Yau for deriving the third order bound was intorduced by Calabi in his ingenuous calculation in the case of real Monge-Amp`ere equation, and If I'm not mistaken, Yau's method for bounding $S$ using  the maximum principle to was due to Nirenberg.
The continuity method itself was already classical, know, probably since Sergej Bernstein's work.
