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There are two separate places where ingenious uses of gauge transformations simplify the analysis of Ricci flow considerably. 

The Deturck trick is a way to break the diffeomorphism invariance of the Ricci flow (or the prescribed Ricci problem) by fixing a gauge to remove the degeneracy of the symbol. This allows for a much simpler proof of the existence and uniqueness of the flow without using the Nash-Moser inverse function theorem.

The Uhlenbeck trick is another gauge transformation which plays an important role in the analysis. Heuristically, one evolves an orthonormal frame in a particular way along Ricci flow. There's also a conceptual approach to this which fixes a background metric on an abstract vector bundle and evolves a bundle isomorphism. Doing so greatly simplifies the reaction terms for the curvature evolution, and plays an important role in many of the "null-vector condition" calculations.

Is it merely a coincidence that gauge transformations show up in both places, or is there some deeper connection at work here?

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    $\begingroup$ Both are very clever tricks. My impression is that Uhlenbeck introduced (at least) two "tricks" in her work on Yang-Mills. One is to break gauge invariance by fixing a gauge such as the Coulomb gauge. DeTurck's trick is equivalent to this in the Ricci flow setting. Then there is the Uhlenbeck trivialization trick, which means write everything with respect to a cleverly chosen frame. My impression is that she used this in her work on Yang-Mills and showed it could also simplify calculations in the Ricci flow setting, Based on all this, I would say that the two tricks are different. $\endgroup$
    – Deane Yang
    Commented Dec 28, 2020 at 22:44
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    $\begingroup$ I don't think Uhlenbeck ever worked on the Ricci flow. I think she showed her trick to Hamilton, and he used it. $\endgroup$
    – Deane Yang
    Commented Dec 28, 2020 at 22:47
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    $\begingroup$ @DeaneYang That's correct. Hamilton attributed it to Uhlenbeck in "Three-manifolds with positive Ricci curvature," which is presumably how it got the name. $\endgroup$
    – Gabe K
    Commented Dec 28, 2020 at 22:57
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    $\begingroup$ My recollection also is that Uhlenbeck claims she did not know whether her trick would help or not. When she saw the awful calculations Hamilton was trying to do (everybody was awestruck by how he managed to navigate through such long and messy equations), she suggested he try using her trick. $\endgroup$
    – Deane Yang
    Commented Dec 28, 2020 at 23:10
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    $\begingroup$ It may be worth noting that the DeTurck trick is nonlinear, amounting to a coupling with the harmonic map heat flow, and the Uhlenbeck trick is from a linear ODE, resulting in a one-parameter family of self-maps of the set of bases of a tangent space. So arguably one shouldn't expect any relation. $\endgroup$
    – Quarto Bendir
    Commented Dec 30, 2020 at 10:39

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