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Quarto Bendir's user avatar
Quarto Bendir's user avatar
Quarto Bendir's user avatar
Quarto Bendir
  • Member for 4 years, 8 months
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Bahri's Five Gaps in Mathematics paper
@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
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Top journals in mathematical analysis
CPAM and GAFA are equally top-rank journals for analysis
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Low boundary of $\mathcal W$ function
See chapter 6 pages 234-235 of Chow et al doi.org/10.1090/surv/135
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How to show the upperbound of the Ricci tensor preserved on 3 manifold
Is your confusion about the sentence "The entry of $N_{ij}$ in question... " on p.282?
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The Ricci curvature is bounded below by scalar curvature
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
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Influence of Yau's solution to the Calabi Conjecture on the field of PDEs
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.
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Influence of Yau's solution to the Calabi Conjecture on the field of PDEs
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
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Short time existence for fully nonlinear parabolic equations
Possibly Lieberman's book "Second order parabolic differential equations" is where you want to look
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Short time existence for fully nonlinear parabolic equations
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
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Short time existence for fully nonlinear parabolic equations
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
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