2009/08/06 by A. Caminha, Caminha, A., P. Sousa +3 · 1 citation
Engineering · Materials Science · Mathematics · #53C42 #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Polymer Foaming and Composites
paper · pdf · doi:10.48550/arxiv.0908.0786
openalex publication_date 2009/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study foliations of space forms by complete hypersurfaces, under some mild conditions on its higher order mean curvatures. In particular, in Euclidean space we obtain a Bernstein-type theorem for graphs whose mean and scalar curvature do not change sign but may otherwise be nonconstant. We also establish the nonexistence of foliations of the standard sphere whose leaves are complete and have constant scalar curvature, thus extending a theorem of Barbosa, Kenmotsu and Oshikiri. For the more general case of \em r-minimal foliations of the Euclidean space, possibly with a singular set, we are able to invoke a theorem of Ferus to give conditions under which the nonsigular leaves are foliated by hyperplanes.