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Recently, I have been studying the Carleman Similiarity Principle, which is used to study the regularity and unique continuation of J-holomorphic curves. Roughly, one takes a solution $ u $ of a certain PDE (a la Cauchy-Riemann) and tries to find a tranformation $ \Phi $ and a holomorphic $ \sigma $ with $ u = \Phi \sigma$.

Now, the construction starts as follows: it is obvious that, given a family of almost complex structures $ J \in W^{1,p}(B, R^{2n \times 2n}), J^2 = -Id$, where $B \subset \mathbb{C}$ is a small ball around the origin, one can find a trivializing transformation $\Psi \in W^{1,p}(B, Hom_{\mathbb{R}}(\mathbb{C}^n, \mathbb{R}^{2n})) $, so that $\Psi^{-1}(z) J(z) \Psi(z) = J_0$ (here $J_0$ is the standard complex structure).

Is this step really straightforward?

It seems, this should be true by finding a conjugation $\Psi$ at $z = 0$ and then using the standard Implicit Function Theorem, provided that $J \in C^1$. However, here the regularity is $W^{1,p}$ (in particular, $C^0$ by Morrey's theorem).

Geometrically, I understand that the almost complex structure on the trivial bundle $R^{2n} \times B \rightarrow B$ is transformed to the canonical constant $J_0$.

Thanks in advance (and sorry, if the question is indeed trivial).

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    $\begingroup$ If you are referring to the proof in the big McDuff-Salamon book, they state at the beginning of that section (2.3) an assumption for J to have weakest possible smoothness, C^1. I think this assumption is meant to permeate through. $\endgroup$
    – Chris Gerig
    Commented Oct 15, 2016 at 4:45
  • $\begingroup$ Thanks for the comment. Yes, I am talking exactly about the McDuff-Salamon book you mentioned. However, it seems they don't wish to increase the regularity: they take $J(z) := J(u(z))$, where $u \in W^{1,p}$. So, I believe the statement should hold for $J$ as mentioned above. Anyhow, how about taking a section of the projection $\Psi \mapsto \Psi^{-1} J_0 \Psi $ and then precomposing it with $J$? $\endgroup$
    – Boggie Georgiev
    Commented Oct 15, 2016 at 10:06

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It seems one can use the fact that $GL(2n, \mathbb{R})$ acts transitively on the space of almost complex structure $C$ by conjugation. One gets a map $GL(2n, \mathbb{R}) \rightarrow C$ of maximal rank. Thus, one can find locally a smooth section $C \rightarrow GL(2n, \mathbb{R}) $. Precomposing this section with $J(z)$ gives the needed canonization.

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