Gluing constructions amongst constant mean curvature hypersurfaces in $${\mathbb {S}^{n+1}}$$

Abstract

Four constructions of constant mean curvature (CMC) hypersurfaces in S n+1 are given, which should be considered analogues of 'classical' constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all the values of the mean curvature. Second, a hypersurface is constructed which consists of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached to each other by symmetrically positioned Delaunay-like 'arms'. This last result extends Butscher and Pacard's doubling construction for generalized Clifford tori of small mean curvature.

Constant mean curvature hypersurfaces in $${{\mathbb S}^{n+1}}$$ by gluing spherical building blocks

“…Suppose that balancing condition (2) holds and also that the mapping between finitedimensional vector spaces which takes small displacements of the geodesics forming # ,τ to the quantity given by the left hand side of (2) has full rank. If τ is sufficiently small, theñ ,τ can be perturbed into an exactly CMC hypersurface ,τ .…”

“…The proof of this theorem will follow broadly the same lines as Main Theorem 2 in Butscher's paper [2]. That is, it will be shown that the partial differential equation for the graphing function whose solution gives a CMC perturbation of˜ ,τ can be solved up to a error term belonging to a finite dimensional obstruction space spanned by the approximate Jacobi fields of˜ ,τ (as explained more fully in [2] and in the proof below).…”

“…Indeed, there are obstructions for solving the CMC equation on an arbitrary initial configuration in S n+1 that are analogous to the obstructions appearing in Euclidean space; and just as in the Euclidean setting, when certain global geometric conditions are met, the obstructions disappear. However, the geometric conditions found in [2][3][4] for S n+1 constructions, though close analogues of the conditions identified by Kapouleas for Euclidean space, appear to be somewhat stronger. This is to be expected since S n+1 is compact and the additional requirement that the initial configurations must close should have ramifications in the analysis of the CMC equation.…”

“…In Butscher's and Butscher-Pacard's work [2][3][4], the gluing technique for constructing CMC hypersurfaces has been successfully adapted to work in the compact ambient manifold S n+1 . In these papers, the CMC building blocks of the sphere-namely the hyperspheres obtained by intersecting S n+1 with hyperplanes and the product spheres of the form S p (cos(α))× S q (sin(α)) for α ∈ (0, π/2) called the generalized Clifford tori-are configured in a variety of different ways, glued together using small embedded catenoidal necks, and perturbed into CMC hypersurfaces.…”

“…Indeed, the totality of local symmetry conditions imposed by force balancing and the fact that CMC hypersurfaces in S n+1 must close seems to force a degree of global symmetry on the initial configuration; and the methods developed in [2] do not seem to apply perfectly to initial configurations with small symmetry groups. However, this paper will show that it is quite possible to develop a gluing technique that is applicable to initial configurations with less symmetry.…”