Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
@MoisheKohan Neither Kleiner-Lott nor Cao-Zhu's expositions covered this part of Perelman's work to any extent whatsoever, Morgan-Tian's was the only one to do so
Are you suggesting that Hamilton's proof could be simplified? What you say is correct, and present in Hamilton's paper in the two sentences "Now if $R_{ij}\geq\varepsilon Rg_{ij}$ ... so $\mu+\nu-2\lambda\geq 0$." on p.281
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.
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
Also see Huisken and Polden's article "Geometric evolution equations for hypersurfaces" for a careful treatment of quasilinear evolution equations of arbitrary order on a manifold
Whenever you have any kind of estimates for the problem as linearized around an open set of functions, it should be automatically possible to appeal to Hamilton's Nash-Moser theorem. And then the nonlinear estimates automatically follow. But this must be an overcomplication for the specific problem you are asking