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.