Foliation of 3-dimensional space forms by surfaces with constant mean curvature

Abstract

Let Y represent a 3-dimensional complete simply-connected space form. We study C2-foliations g of Y by leaves with the same constant mean curvature. We prove that if the curvature of Y is positive such foliations can not exist, When Y is the Euclidean space then such a foliation must consist of parallel planes. When Y is the hyperbolic space, if we further assume that the mean curvature satisfies H > l, then F must be a foliation by horospheres. These results are still true if F is a fol~atlon of an open set U of Y and if we further assume that the leaves are complete and orientable. We observe that on hyperbolic space there are examples of nontrlvial foliations of open sets by complete surfaces with the same constant mean curvature 0 < H < I. One example can be obtained from the l-parameter family of catenoids studied by do Carmo and Dajczer ~D] and by Gomes ~]. To prove the results, we consider a codimension-one foliation of an orientable Riemannian manifold, whose leaves are orientable and have the same constant mean curvature, and first show that its leaves are strongly stable in the sense defined in ~CE]. We then apply the classification theorem for complete stable surfaces of a 3-dimensional space form proved in ~CE] and ~].