Complete Foliations of Space Forms by Hypersurfaces

TL;DR

This paper investigates the structure of foliations by complete hypersurfaces in space forms, establishing new nonexistence and classification results under curvature conditions, including Bernstein-type theorems and hyperplane foliations.

Contribution

It extends classical results by providing new conditions for the existence and nonexistence of certain hypersurface foliations in space forms, including Euclidean and spherical cases.

Findings

Bernstein-type theorem for graphs with sign-changing mean and scalar curvature in Euclidean space.

Nonexistence of complete, constant scalar curvature foliations in the standard sphere.

Conditions under which r-minimal foliations in Euclidean space are foliated by hyperplanes.

Abstract

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.