Caccioppoli’s inequalities on constant mean curvature hypersurfaces in Riemannian manifolds (original) (raw)

Unicity of constant mean curvature hypersurfaces in some riemannian manifolds

Indiana University Mathematics Journal, 1999

We study compact hypersurfaces of constant mean curvature immersed into a wide class of Riemannian manifolds having a foliation whose leaves are umbilical hypersurfaces. In this situation we will obtain, among other things, the analogues to the classical Jellett-Liebmann and Alexandrov theorems for hypersurfaces in Euclidean space.

Stable constant mean curvature hypersurfaces in ℝ n+1 and ℍ n+1(−1)

Bulletin of the Brazilian Mathematical Society, New Series, 2005

In this paper, we show that all complete stable hypersurfaces in R n+1 (or H n+1 (−1)) (n = 3, 4, 5) with constant mean curvature H > 0 (or H > 1, respectively) and finite L 2 norm of traceless second fundamental form are compact geodesic spheres.

Some geometric properties of hypersurfaces with constant rrr-mean curvature in Euclidean space

Proceedings of the American Mathematical Society, 2010

Let f : M → R m+1 be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors in [4] to analyze the stability of the differential operator L r associated with the r-th Newton tensor of f . This appears in the Jacobi operator for the variational problem of minimizing the r-mean curvature H r . Two natural applications are found. The first one ensures that, under a mild condition on the integral of H r over geodesic spheres, the Gauss map meets each equator of S m infinitely many times. The second one deals with hypersurfaces with zero (r + 1)-mean curvature. Under similar growth assumptions, we prove that the affine tangent spaces f * T p M , p ∈ M , fill the whole R m+1 .

Higher order mean curvature estimates for bounded complete hypersurfaces

Nonlinear Analysis: Theory, Methods & Applications, 2013

We obtain sharp estimates involving the mean curvatures of higher order of a complete bounded hypersurface immersed in a complete Riemannian manifold. Similar results are also given for complete spacelike hypersurfaces in Lorentzian ambient spaces. * This work was partially supported by MINECO (Ministerio de Economía y Competitividad) and FEDER (Fondo Europeo de Desarrollo Regional) project MTM2012-34037 and Fundación Séneca project 04540/GERM/06, Spain.

Embedded Constant Mean Curvature Hypersurfaces on Spheres

Asian Journal of Mathematics, 2010

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) √ n−1 n √ m 2 −1 , 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.

Hypersurfaces in Non-Flat Pseudo-Riemannian Space Forms Satisfying a Linear Condition in the Linearized Operator of a Higher Order Mean Curvature

Taiwanese Journal of Mathematics, 2013

We study hypersurfaces either in the pseudo-Riemannian De Sitter space S n+1 t ⊂ R n+2 t or in the pseudo-Riemannian anti De Sitter space H n+1 t ⊂ R n+2 t+1 whose position vector ψ satisfies the condition L k ψ = Aψ + b, where L k is the linearized operator of the (k+1)-th mean curvature of the hypersurface, for a fixed k = 0, . . ., n−1, A is an (n+2)×(n+2) constant matrix and b is a constant vector in the corresponding pseudo-Euclidean space. For every k, we prove that when H k is constant, the only hypersurfaces satisfying that condition are hypersurfaces with zero (k +1)-th mean curvature and constant k-th mean curvature, open pieces of a totally umbilical hypersurface in S n+1 t (S n t−1 (r), r > 1; S n t (r), 0 < r < 1;

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 R n+1 , n ≥ 3, must have only one end. Any complete noncompact weakly stable CMC H-hypersurface in the hyperbolic space H n+1 , n = 3, 4, with H 2 ≥ 10 9 , 7 4 , respectively, has only one end. * Supported by CNPq of Brazil † Supported by CAPES and CNPq of Brazil.