Large outlying stable constant mean curvature spheres in initial data sets

Constant mean curvature hypersurfaces foliated by spheres ∗

We ask when a constant mean curvature n-submanifold foliated by spheres in one of the Euclidean, hyperbolic and Lorentz–Minkowski spaces (En+1, Hn+1 or Ln+1), is a hypersurface of revolution. In En+1 and Ln+1 we will assume that the spheres lie in parallel hyperplanes and in the case of hyperbolic space Hn+1, the spheres will be contained in parallel horospheres. Finally, Riemann examples in L3...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Stable constant mean curvature hypersurfaces in some Riemannian manifolds

We determine all stable constant mean curvature hypersurfaces in a wide class of complete Riemannian manifolds having a foliation whose leaves are umbilical hypersurfaces. Among the consequences of this analysis we obtain all the stable constant mean curvature hypersurfaces in many nonsimply connected hyperbolic space forms. Mathematics Subject Classification (1991). Primary 53A10; Secondary 49...

Stable periodic constant mean curvature surfaces and mesoscopic phase separation

We give a first comprehensive description of the stable solutions of the periodic isoperimetric problem in the case of lattice symmetry. This result is intended to elucidate the geometry of certain sophisticated interfaces appearing in mesoscale phase separation phenomena. We prove that closed, stable, constant mean curvature surfaces in R3/Γ , Γ ⊂ R3 being a discrete subgroup of translations w...

The structure of stable constant mean curvature hypersufaces

We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature hypersurfaces in space forms. In particular, a complete oriented weakly stable minimal hypersurface in Rn+1, n ≥ 3, must have only one end. Any complete noncompact w...