Folheações por hipersuperfícies de curvatura média constante

Abstract

n this paper, we work showing results aiming classify foliations of codimension-one in Riemannian manifolds whose leaves have constant mean curvature. The main result is the theorem by Barbosa-Kenmotsu-Oshikiri([3]). Theorem: LetM be a compact Riemannian manifold with nonnegative Ricci curvature e F, a codimensiononeC3-foliation of M whose leaves have constant mean curvature. The any leaf of F is totally geodesic submanifold of M. Futhermore M is locally a Riemannian product of a leaf of F and a normal curve,and the Ricci curvature in the direction normal to the leaves is zero. The previous result can not be extended for the case where M is not compact. A foliation counterexample can be built from a function f that does not satisfy the Bernstein’s conjecture. At the end, they are present recent results about the boarded problems and a proof of the Heinz-Chern inequality.