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 bm,n = (m 2 −2), there exists a non isoparametric, compact embedded hypersurface in S n+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( π m ), 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 H n+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 S n+1 whose cones in R n+2 are stable. Also, we prove that the stability index of every non-isoparametric minimal hypersurface with two principal curvatures in S n+1 exceeds n + 3.

Serrin’s overdetermined problem on the sphere

“…By the same technique, Alexandrov proved in [3] that any closed embedded CM C-hypersurface contained in a hemisphere of S N is a geodesic sphere. An explicit family of embedded CM C-hypersurfaces in S N with nonconstant principal curvatures has been found by Perdomo in [26] in the case N ≥ 3. These hypersurfaces seem somewhat related to the Serrin domains given by Theorem 1.1, although they bound a tubular neighborhood of S 1 and not of S N −1 .…”

Delaunay-Type Hypersurfaces in Cohomogeneity One Manifolds

“…Shortly after the announcement of the proof of the Lawson Conjecture, Andrews and Li [6] announced a full classification of embedded CMC tori in S 3 , proving a conjecture of Pinkall and Sterling that asserts that any such embedding is rotationally symmetric, see also [41]. This reduced the classification to a known case, see [25,40]. The above minimal Clifford torus is an example of Delaunay surface.…”

“…The study of CMC hypersurfaces on spheres is a classic subject in differential geometry, that saw important developments in recent years, see e.g. [4,6,13,40,41]. A major contribution to the area was recently given by Brendle [13] with a proof of the Lawson Conjecture, that states that the minimal Clifford torus…”

Two-point functions and their applications in geometry

“…This leads to the result that every constant mean curvature torus in S 3 which is immersed in the sense of Alexandrov must be rotationally symmetric; see [4] and [9] for details. Moreover, the classification of constant mean curvature tori in S 3 with rotational symmetry can be reduced to the analysis of an ODE (see [46]). It turns out that the class of surfaces which are immersed in the sense of Alexandrov is quite natural in this context; in particular, there is a large class of examples which are immersed in the sense of Alexandrov, but fail to be embedded.…”