Uhlenbeck's theorem novelty This link provides a short introduction to the contributions of Uhlenbeck about regular gauge fixing. However, I feel quite puzzled about it and I do not understand the real novelty apported by this work. More concisely, it would be helpful for me to understand

*

*which is the transcendence and novelty of the local Coulomb gauge theorem (theorem 3.2), in case it can be shortly explained, and what it is useful for.


*which relation has this result with physics (I have heard in the web that the so-called $\epsilon$ plays some Planck's constant role, and that everything may be translated into quantum physics, or at least physics world).
Any suggestion for those two questions will be welcomed.
 A: Denote by $A$ the connection and  by $F_A$ its curvature.    Then
$$dA=F_A-A\wedge A. $$
If $A$ is in Coulomb gauge we have an additional equation
$$d^*A=0. $$
The advantage is that the operator $d\oplus d^*$ is elliptic and now we  have an equation of the form
$$ (d\oplus d^*)A= \mbox{something}. $$
Elliptic theory  allows us to convert  bounds on  "something'' into bounds on $A$.    Then, the bounds on  $A$ can be converted into compactness results using   standard    compactness results  in Sobolev spaces.
Finding a  local Coulomb gauge on a region $D$ is possible as long as  the "energy"  $\Vert F_A\Vert_{L^2(D)}$  is smaller than $<\epsilon$, where $\epsilon$ is related to  the second Chern number  of a principal  $G$-bundle over $S^4$, the conformal compactification of $\mathbb{R}^4$.  The energy of an instanton on $S^4$ is equal, up to a universal constant, to  the second Chern number which is an integer. You can regard  this as a quantization result, stating  that the energy of an instanton is an integral multiple of a universal constant. (If my memory serves me right this constant is $4\pi^2$, give or take a factor of $2$.)
