# 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 few complex geometers have told me that nobody had previously figured out Moser iteration for anything but linear PDE, but I think they were unaware of Serrin or Trudinger's papers, and likely others as well. As I understand it, Yau's paper is an impressive synthesis of existing techniques does not introduce fundamentally new analysis. Knowing how to formulate and carry out such analysis on manifolds is much easier today than then (with this as a major step) but that is perhaps a separate question Mar 16, 2022 at 14:21

## 1 Answer

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.

• This is all correct but I think it should be clarified that the Pogorelov and Calabi computations cannot be automatically carried over, and need to be adapted to the complex Monge-Ampere equation and the setting of plurisubharmonicity instead of convexity. It is not direct to see that this can be done. Also, along with Calabi and Pogorelov for the third and second order estimates, it should be noted that the method for the zeroth order estimate has its origin in Moser's iteration technique from 1960-61. Mar 16, 2022 at 14:30