# Complex manifold with boundary

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.

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

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.

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.

• 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? – André Henriques Apr 30 '19 at 22:08
• 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? – André Henriques Apr 30 '19 at 22:18
• 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. – Igor Khavkine Apr 30 '19 at 23:32