Foliations by constant mean curvature tubes

Abstract

Constant mean curvature hypersurfaces constitute a very important class of submanifolds in a compact Riemannian manifold (M, g). In this paper we are interested in families of such submanifolds, with mean curvature varying from one member of the family to another, which form (partial) foliations and which ‘condense’ to a submanifold Γ ⊂ M of codimension greater than 1. Our main results concern the existence of such families and, conversely, the geometric nature of the submanifolds Γ to which such families can condense. The simplest case, where Γ is a point, was considered by Ye a decade ago, [12], [13]. He proved that if p ∈ M is a nondegenerate critical point of the scalar curvature function Rg, then there exists a neighborhood U 3 p such that U \{p} is foliated by constant mean curvature (for short CMC) spheres; in fact, the members of this family are small perturbations of the geodesic spheres of radius ρ, 0 < ρ < ρ0, and hence they have mean curvatures H = 1/ρ → ∞. Moreover, this foliation is essentially unique. Conversely, if a neighbourhood of p admits such a foliation, then necessarily ∇Rg|p = 0. In very closely related work, Ye [14], and by quite different methods (using inverse mean curvature flow) Huisken and Yau [4], proved the existence of a unique foliation by CMC spheres near infinity in an asymptotically flat manifold (of nonnegative scalar curvature); this is of interest in general relativity. In this paper we study the existence of families of CMC hypersurfaces which converge to a (closed, embedded) submanifold Γ ⊂ M, particularly in the case ` = 1. Define the geodesic tube