Embedded constant mean curvature hypersurfaces on spheres

Abstract

Let m ≥ 2 and n ≥ 2 be any pair of integers. In this paper we prove that if H lies between cot(π/m) and there exists a non isoparametric, compact embedded hypersurface in Sn+1 with constant mean curvature H that admits O(n) × Zm in its group of isometries. These hypersurfaces therefore have exactly 2 principal curvatures. When m = 2 and H is close to the boundary value 0 = cot(π/2), such a hypersurface looks like two very close n-dimensional spheres with two catenoid necks attached, similar to constructions made by Kapouleas. When m > 2 and H is close to cot(m/π), it looks like a necklace made out of m spheres with m + 1 catenoid necks attached, similar to constructions made by Butscher and Pacard. In general, when H is close to bm,n the hypersurface is close to an isoparametric hypersurface with the same mean curvature. For hyperbolic spaces we prove that every H ≥ 0 can be realized as the mean curvature of an embedded CMC hypersurface in Hn+1. Moreover we prove that when H > 1 this hypersurface admits O(n)×Z in its group of isometries. As a corollary of the properties we prove for these hypersurfaces, we construct, for any n ≥ 6, non-isoparametric compact minimal hypersurfaces in Sn+1 whose cones in Rn+2 are stable. Also, we prove that the stability index of every non-isoparametric minimal hypersurface with two principal curvatures in Sn+1 exceeds n + 3. © 2010 International Press.