Let $H\subseteq\mathbb C^n$ be a smooth co-oriented real codimension one hypersurface. If $H$ is weakly pseudo-convex, then holomorphic maps $u:\Delta\to\mathbb C^n$ ($\Delta$ denotes the unit disk) satisfy the following maximum principle with respect to $H$:

If $u(0)\in H$ and $u$ maps a neighborhood of $0$ to the "inside" of $H$, then in fact $u$ maps a neighborhood of $0$ to $H$.

Moreover, my understanding is that this result is sharp: if $H$ is not weakly pseudo-convex, then there are maps $u$ violating the maximum principle.

Question: Is there a sharp maximum principle for pseudo-holomorphic maps $u:\Delta\to(X^{2n},J)$ where $J$ is a (not necessarily integrable) almost complex structure and $H\subseteq X$ is as before?

I know of various sufficient conditions on $H$ which imply a maximum principle, however they generally require rather special assumptions on the local geometry of $J$ (such as the existence of a Liouville $1$-form) which seem far from the most general possible. It would be very nice to know what the "right" assumptions are here.

  • $\begingroup$ A. Sukhov is the man to talk to. $\endgroup$ – Ben McKay Mar 5 '17 at 8:00
  • $\begingroup$ I think that you can take any Hermitian metric and define a Levi form for the boundary of an almost complex manifold, and then prove that its positive definiteness doesn't depend on the almost complex structure. Then you can prove something like the same result as for the complex case. I did something like this (without the Hermitian metric) in my thesis arxiv.org/abs/math/0101017 $\endgroup$ – Ben McKay Mar 5 '17 at 9:40

You may already know this, and this is not the most general statement, probably, but it does work for all almost complex structures (integrable or not):

First, some notation: Using the almost complex structure $J$ on $X$, split the exterior derivative into graded pieces $d^{s,t}:\Omega^{p,q}(X)\to\Omega^{p+s,q+t}(X)$, where $(s,t)\in\{(2,-1),(1,0),(0,1),(-1,2)\}$. Of course, $d^{2,-1}$ and $d^{-1,2}$ are tensorial (i.e., zeroth order linear differential operators) and vanish when $J$ is integrable. Maintain the standard abbreviations $d^{1,0}=\partial$ and $d^{0,1}=\overline\partial$. [Note that, because pseudo-holomorphic mappings preserve bi-degree (and, of course, 'commute' with the exterior derivative), these operators commute with pullback via pseudo-holomorphic maps.]

Given a (say, smooth) function $f:X\to\mathbb{R}$ define the Levi form of $f$ to be $$ {\mathsf L}(f) = i\,\partial\overline{\partial}\,f \in \Omega^{1,1}(X). $$ Then ${\mathsf L}(f)$ is a real-valued $(1,1)$-form on $X$. One says that $f$ is plurisubharmonic on $X$ if ${\mathsf L}(f)\ge0$ (in the sense that it is nonnegative on all (oriented) $1$-dimensional complex subspaces of $TX$).

Proposition: Let $f:X\to\mathbb{R}$ be plurisubharmonic. If $u:\Delta\to X$ is $J$-holomorphic and $f\bigl(u(z)\bigr)\le f\bigl(u(0)\bigr)$ for all $z\in\Delta$, then $f\bigl(u(z)\bigr) =f\bigl(u(0)\bigr)$ for all $z\in \Delta$.

The relation with the OP's question is this: If $X\subset\mathbb{C}^n$ and $c$ is a regular value of $f:X\to\mathbb{R}$, where $f$ is plurisubharmonic, then $H_c=f^{-1}(c)$ is weakly pseudo-convex.

It's not clear to me that a smooth weakly pseudo-convex hypersurface in $X$ is locally a level set of a plurisubharmonic function $f$, but it seems plausible. (See the comment below, which asserts that this is true in the integrable case, by a result of Diederich-Fornæss.)

The proof of the above proposition follows from the (usual) maximum principle on $\Delta$ and the formula $$ \frac{\partial^2(f{\circ}u)}{\partial z\,\partial \overline z} = \frac1i \,u^*\bigl({\mathsf L}(f)\bigr)\left(\frac{\partial}{\partial z},\frac{\partial}{\partial \overline{z}}\right) \ge0, $$ which is a consequence of the fact that the Levi form operator commutes with pullback under pseudo-holomorphic maps. (Note, in particular, that the Nijnhuis tensor does not affect the formula.)

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    $\begingroup$ In the case of integrable J, a C^2 weakly pseudo-convex hypersurface seems to be always locally defined by a weakly pseudo-convex function f by a result of Diederich–Fornæss. $\endgroup$ – user_1789 Mar 5 '17 at 16:55
  • $\begingroup$ @user_1789: Thanks for this comment. Do you have a specific reference for the result of Diederich-Fornæss? That is interesting, and it may be generalizable to the non-integrable case. $\endgroup$ – Robert Bryant Mar 6 '17 at 9:45
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    $\begingroup$ A reference to D-F is contained in a paper of Range. He gives an alternative proof of essentially the same result, which might be easier to adapt. There may well be more recent relevant work of which I am unaware. $\endgroup$ – user_1789 Mar 6 '17 at 10:53
  • $\begingroup$ @user_1789: Thanks! I'll have a look. $\endgroup$ – Robert Bryant Mar 6 '17 at 12:06
  • $\begingroup$ Actually, it seems that if J is C^2, such that the Levi form is C^1 (and such that there is a strictly plurisubharmonic function $-d(dr \circ J)$ in a neighborhood of a point p, where $r=dist(p,\cdot)^2$), Range's proof might carry over word by word to give the desired result. $\endgroup$ – user_1789 Mar 6 '17 at 12:45

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