Is there a maximum principle for CR functions over domains inside CR manifolds?

Question

I am new to this area and I am a bit confused by the literature. Is there a strong maximum principle for CR functions over domains in a CR manifold, please? If so, could someone please state it (together with its hypotheses etc.) and provide a reference to a proof (or maybe just prove it if it is not too long)? In the mean time, I will keep on digging in the literature, but in any case, it may be of interest to others too.

Edit 1: it looks like the answer is no, according to ex. 3.2.5 in CR functions (LibreText Mathematics), by Jiří Lebl. Actually, this makes sense, because for example, in the simplest case where our CR manifold is a real hypersurface in $\mathbb{C}^{n+1}$ given by $z_{n+1} - \bar{z}_{n+1} = 0$, then being a CR function is, if I understand the definition well, just being holomorphic in the $z_i$, for $i = 1, \ldots, n$, but it says nothing about how the function depends on $x_{n+1}$ ($x_{n+1}$ is the real part of $z_{n+1}$).

But in my case, the function is also real analytic. So allow me to ask if there is a strong maximum principle for real analytic CR functions on a domain inside a CR manifold. I'd also be happy to assume that (if it helps), at each point $p \in M$, the $T_p^{(1,0)}$ bundle of $M$ has complex rank $n$, where the dimension of $M$ is $2n+1$ (so that, in some sense, $M$ is like a real hypersurface inside some complex $n+1$ dimensional manifold).

Answer 1

It's not really about real-analytic or smooth. It is really about the CR structure of the manifold. In your case, the manifold is given by $\operatorname{Im} z_{n+1} = 0$, so any CR function is just a function holomorphic in the first $n$ variables, as you noted. Hence, any CR function on your manifold that achieves a maximum will be constant along that "leaf", that is, if $f$ is your CR function on your manifold and achieves a maximum at the origin, then $z' \mapsto f(z',0)$ (where $z' \in {\mathbb{C}}^n$) is constant, but that's the best you can get whether real-analytic or not.

Try to find a counterexample to a strong maximum principle by considering the sphere, which is a wonderful CR manifold. Your counterexample will be a polynomial, so definitely real-analytic. It will be the restriction of an entire function.

On the other hand for a manifold such as the Lewy surface, $\operatorname{Im} z_3 = |z_1|^2-|z_2|^2$, you can prove a maximum principle. For this, look at the chapter in the book that talks about extension and attaching discs. Then note that the attached discs will force the extended holomorphic function to have a maximum at the origin and the standard maximum principle for holomorphic functions applies. To see this, consider an analytic disc attached to the manifold, suppose your function is already extended as a holomorphic function, and note that the one-dimensional maximum principle tells you that the function inside the disc is bounded by its values on the boundary, which are the values of your CR function on the CR manifold.