On foliations of \({\mathbb{R}}^{n+1}\) by minimal hypersurfaces (Q1081128)

C\({}^ 0\)-foliations of \(R^{n+1}\) by minimal hypersurfaces are considered. The study is motivated by the Bernstein theorem [\textit{E. Bombieri}, \textit{E. De Giorgi} and \textit{E. Giusti}, Invent. Math. 7, 243- 268 (1969; Zbl 0183.259)]. The main result is local: Any codimension-one \(C^ 0\)-foliation of an open set \(U\subset R^{n+1}\) is locally Lipschitz provided that all the leaves are minimal. (Note, that there are minimal foliations like these which are not \(C^ 1 !)\) The following results are global: (1) If \({\mathcal F}\) is a \(C^ k\)-foliation (k\(\geq 0)\) of \(R^{n+1}\) by proper minimal hypersurfaces, then the space of leaves of \({\mathcal F}\) is \(C^ k\)-diffeomorphic to R. (2) If \({\mathcal F}\) as above admits a leaf which is asymptotically regular, then \({\mathcal F}=\{L\times \{t\}\); \(t\in R\}\) and L is contractible.