HYPERSURFACES WITH CONSTANT k-TH MEAN CURVATURE AND TWO DISTINCT PRINCIPAL CURVATURES IN SPHERES

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

Rigidity of minimal hypersurfaces of spheres with two principal curvatures

Let ν be a unit normal vector field along M . Notice that ν : M −→ S satisfies that 〈ν(m),m〉 = 0. For any tangent vector v ∈ TmM , m ∈ M , the shape operator A is given by A(v) = −∇̄vν, where ∇̄ denotes the Levi Civita connection in S. For every m ∈ M , A(m) defines a linear symmetric transformation from TmM to TmM ; the eigenvalues of this transformation are known as the principal curvatures of ...

$L_1$-Biharmonic Hypersurfaces in Euclidean Spaces with Three Distinct Principal Curvatures

Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...

Stability of Hypersurfaces with Constant R-th Anisotropic Mean Curvature

Given a positive function F on S which satisfies a convexity condition, we define the r-th anisotropic mean curvature function H r for hypersurfaces in R n+1 which is a generalization of the usual r-th mean curvature function. Let X : M → R be an n-dimensional closed hypersurface with H r+1 =constant, for some r with 0 ≤ r ≤ n− 1, which is a critical point for a variational problem. We show tha...