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Let $C(p_1, p_2; p_3, p_4)$ denote the cross-ratio of the $4$ points $p_i$, for $i = 1, \ldots, 4$, thought of as a holomorphic function on $$ \Omega = \{ (p_1, p_2, p_3, p_4) \in \mathbb{C}P^1 \times \mathbb{C}P^1 \times \mathbb{C}P^1 \times \mathbb{C}P^1; \text{the $p_i$ are all distinct} \}.$$

Given a real submanifold $M \subseteq \Omega$ (Edit: the dimension of $M$ is allowed to possibly be smaller than $\operatorname{dim}(\Omega) = 8$) and a domain $D \subseteq M$, does the strong Hopf maximum principle apply to the real part of $C$ on the domain $D$?

If this is not always true, can we find sufficient conditions on $M$ and $D$ for the maximum principle to apply?

I think some statement like the above should be true, because not only is $C$ holomorphic, but it is also affine in each $p_i$, individually.

However, given $M$ and $D$, I am not sure how to construct a kind of Laplacian operator on $D$ (well, I mean a 2nd order linear differential operator of the kind appearing in the strong Hopf maximum principle), which contains the real part of $C$ in its kernel (so that the real part of $C$ is harmonic-like, so to speak).

I am also not sure which domains are allowed. In Hopf's maximum principle, the domain is a subset of $\mathbb{R}^n$, but here we need more general domains, which are subsets of a submanifold $M$ of $\Omega$.

Remark: in reply to one of @Alexandre Eremenko's comments below, I point out that $M$ is a subset of $\Omega$ and $\Omega$ was defined so that we avoid tuples of points on $\mathbb{C}P^1$ for which the cross-ratio is $\infty$.

Edit 1: @Alexandre Eremenko pointed out that the maximum principle cannot hold on any domain inside any submanifold, because a submanifold can be very complicated. However, I am still interested in some sufficient conditions on $M$ and $D$ that would guarantee that the (strong) maximum principle holds. I have a feeling that for simple enough domains and submanifolds, we would have a strong maximum principle statement. But I lack a precise statement...

Edit 2: I guess what I am interested in, in terms of $M$ and $D$, is the following case.

$$ M = D = \{ \operatorname{Im}(h_i) = 0 ; \text{ for $i = 1, \ldots, m$} \} \cap \Omega,$$

where $\operatorname{Im}$ denotes the imaginary part and each $h_i$ is a meromorphic function on the product of four copies of $\mathbb{C}P^1$, with properties (degree etc) very similar to those of $C$. I wonder if the maximum principle applies in this special case, assuming that $M = D$ is smooth (no singularities) and connected.

Edit 3: is there a maximum principle over CR manifolds, by any chance? I will dig in the literature. It would possibly help!

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  • $\begingroup$ Since the cross ratio can be $\infty$ you have to choose the domain $D$ where this does not happen. On such a $D$ the real part of the cross ratio is harmonic, even pluriharmonic, thus satisfies the Maximum principle. $\endgroup$
    – Alexandre Eremenko
    Apr 8, 2023 at 0:34
  • $\begingroup$ And this still holds even if $D$ is a domain inside $M$ and the dimension of $M$ is less than the real dimension of $\Omega$ (which is 8), please? This is what I am having trouble with, going to a lower dimensional domain, so to speak. $\endgroup$
    – Malkoun
    Apr 8, 2023 at 0:38
  • $\begingroup$ No. The conclusion argument only applies to complex analytic manifolds. $\endgroup$
    – Alexandre Eremenko
    Apr 8, 2023 at 0:40
  • $\begingroup$ But can't we apply the Hopf maximum principle in this case? It holds in a more general setting, and the cross-ratio has degree at most 1 in each of its arguments... $\endgroup$
    – Malkoun
    Apr 8, 2023 at 0:41
  • $\begingroup$ Certainly no maximum principle can hold for every (real) submanifold. Does not matter that it has degree 1, since the manifold on which you want to restrict can be very complicated. $\endgroup$
    – Alexandre Eremenko
    Apr 8, 2023 at 0:57

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