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My question is of local nature.
Let $$f:\mathbb C^n\to\mathbb R$$ be a $C^\infty$ function that vanishes at $0\in \mathbb C^n$, with non-zero derivative.
Then, around $0\in \mathbb C^n$, $$M:=f^{-1}(0)$$ is a CR manifold. Let me assume that $M$ is the simplest possible kind of CR manifold, namely that it is foliated by real-codimension-one complex submanifolds.

[Equivalently, for those who don't know what CR manifolds are, consider the hyperplane distribution $L:=TM\cap i\cdot TM\subset TM$. I require the distribution $L$ to be integrable, i.e., to come from a (real codimension $1$) foliation of $M$.]

Under the above assumptions, is $f^{-1}\big([0,\infty)\big)$ locally isomorphic to $$\big\{(z_1,...z_n)\in\mathbb C^n\,:\,\mathrm{im}(z_1)\ge 0\big\}?$$

I.e., does there exist a neighbourhood $U\subset f^{-1}([0,\infty))$ of zero and an isomorphism $\varphi:U\to \big\{z\in\mathbb C^n\,:\,\sum|z_i|^2<1,\,\mathrm{im}(z_1)\ge 0\big\}$ which is holomorphic in the interior and smooth all the way to the boundary.

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Perhaps Giuseppe Della Sala's paper might be useful here: https://www.ams.org/journals/proc/2011-139-07/S0002-9939-2010-10746-3/home.html

It precisely deals with the equivalence of smooth Levi-flats. There are examples in the paper

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If $M$ is real analytic then Élie Cartan proved that, in suitable holomorphic coordinates, $M$ is cut out by the imaginary part of $z$. I learned this from the paper https://hal.archives-ouvertes.fr/hal-00459323.

Look for Levi flat hypersurfaces and you will find a lot of literature on the topic.

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I believe you are asking whether the foliation by codimension-1 complex leaves tangent to $L$ can be straightened. It appears that the answer in general is No, as discussed (with examples) in

Freeman, Michael, Local biholomorphic straightening of real submanifolds, Ann. Math. (2) 106, 319-352 (1977). ZBL0372.32005, MR463480.

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  • $\begingroup$ The question of straightening seems indeed related to my question, at least when the manifold $M$ is real analytic (when $M$ is not real analytic, the case of $n=1$ already shows that straightening is not always possible, whereas my question always has a positive answer by the Riemann mapping theorem). Unfortunately, the paper you link doesn't seem to focus on the case when $M$ is a hypersurface, which makes it a bit difficult for me to find the most relevant parts... You claim that there's a counterexamples to my question in that paper. Where is that counterexample? $\endgroup$ – André Henriques Apr 30 '19 at 22:08
  • $\begingroup$ Actually, isn't Thm 3.3(A) of the paper you link a positive answer to my question when $M$ is real analytic? What makes you say that the answer is negative? $\endgroup$ – André Henriques Apr 30 '19 at 22:18
  • $\begingroup$ I must admit that I did get lost in the weeds somewhat when trying the wrap my head around Freeman's results. The negative result seems to be for the general situation where $M$ is locally foliated by $k$-complex-dimensional leaves, where $k$ need not be maximal. The supporting examples are in Sec.5 of the paper. But it is possible that I didn't properly read the caveats about some special cases, like $k$ being maximal as in your question, where no examples are possible. $\endgroup$ – Igor Khavkine Apr 30 '19 at 23:32

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