Complete CMC hypersurfaces in Minkowski \((n+1)\) -space (Q6598516)

The domain of dependence of a space-like hypersurface \(\Sigma\) in Minkowski space \(\mathbb{R}^{n,1}\) is the set of all points \(p\) such that any inextensible causal curve that goes through \(p\) meets \(\Sigma\). The domain of dependence of any entire CMC hypersurface is a regular domain, meaning an open domain obtained as the intersection of at least two future half-spaces bounded by non-parallel null hyperplanes. \N\NThe main result of the paper implies that all regular domains are obtained as domains of dependence of entire CMC hypersurfaces:\N\NTheorem. Given any regular domain \(\mathcal{D}\) in \(\mathbb{R}^{n,1}\) and any \(H > 0\), there exists a unique entire hypersurface \(\Sigma\) of constant mean curvature \(H\) such that the domain of dependence of \(\Sigma\) is \(\mathcal{D}\). Moreover, as \(H\) varies in \((0, +\infty)\), the entire hypersurfaces of constant mean curvature \(H\) analytically foliate \(\mathcal{D}\).