I am interesting in maximal surfaces: space-like surface in Minkowski $\mathbb{R}^{2,1}$ (or De Sitter $dS^3$). Of course space-like implies locally graph and almost all the literature is interested in graph. But local graph doesn't mean global graph as shown by pictures [in this paper](https://www.degruyter.com/downloadpdf/j/anly.1985.5.issue-1-2/anly.1985.5.12.43/anly.1985.5.12.43.pdf). Even before, [Cheng and Yau](https://www.jstor.org/stable/pdf/1970963.pdf?refreqid=excelsior%3A1d5b693664b09fdbe3941c91ffdc8f5e) propose "parametric version"(page 407) of a Bernstein like result of Calabi.


But page 408 they assert that if $p\in \Sigma$ where $\Sigma$ is space like, then $\Sigma\cap C\_p=\{p\}$, where $C\_p$ is the light-cone whose origin is at $p$... considering a flat helix (so space-like) it seems just false, I can find two points joint by a time-like vector on my helix.


The problem is that the whole paper relies on this affirmation, hence I am pretty sure I miss something, but I have think about it, and I still see nothing. Any explanation? Thx in advance.


P.S: is any one as link to the paper of Calabi, Examples of Berstein problems for some nonlinear equations?



maximal surface, parametric approach
The proof given in the Cheng-Yau paper is rather sketchy. However, with the (implicit) assumption that a "closed hypersurface" is meant in the usual sense of manifold without boundary, the result is true. (Note that the statement does *not* require the submanifold to be minimal/maximal.)


**Lemma** If $(M,g)$ is a connected and simply-connected, time-oriented Lorentzian manifold (closed and without boundary), and $S\subset M$ is a closed space-like hypersurface, then $S$ is acausal.


The proof is given as Lemma 45 and Corollary 46 in Chapter 14 of O'Neill's *Semi-Riemannian Geometry*. 


The key part of the proof is the following general fact from intersection theory (I'm stating a very simple version): if $M$ is a closed, simply-connected manifold, and $S$ is a closed, orientable hypersurface, then $S$ separates $M$. See, e.g. <https://math.stackexchange.com/q/1324413/> for similar discussions. 


