Foliations by constant mean curvature tubes

By Rafe Mazzeo, Frank Pacard. 2003

65 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/00967092aa2a0b254eb23eea9d63d454fa9bd65a

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Citations in the corpus

Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains (2005, background): “Remark 1.4 In [ 31 ], the authors consider a non-degenerate closed geodesic in a n-dimensional manifold M. They prove existence of tubular neighborhoods of this geodesic, whose boundary is a hypersurface of constant (and large) mean curvature H. Similarly to our results, resonance phenomena occur, a”
Constant curvature foliations in asymptotically hyperbolic spaces (2007, background): “The papers [15], [14] construct ‘partial’ CMC foliations which collapse to higher dimensional minimal submanifolds.”
Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature (2010, background): “Similar resonance has been observed the problem of building foliations of a neighborhood of a geodesic by CMC tubes considered in [17, 22].”