FOLIATION BY CONSTANT MEAN-CURVATURE SPHERES

Let M be a Riemannian manifold of dimension n + 1 and p is-member-of M. Geodesic spheres around p of small radius constitute a smooth foliation. We shall show that this foliation can be perturbed into a foliation whose leaves are spheres of constant mean curvature, provided that p is a nondegenerate critical point of the scalar curvature function of M. The obtained foliation is actually the unique foliation by constant mean curvature hypersurfaces which is regularly centered at p (Definition 1.1). On the other hand, if p is not a critical point of the scalar curvature function, then there exists no such foliation.