On reduced L2 cohomology of hypersurfaces in spheres with finite total curvature

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...

Rigidity of minimal hypersurfaces of spheres with constant ricci curvature

ABSTRACT: Let M be a compact oriented minimal hypersurface of the unit n-dimensional sphere S. In this paper we will point out that if the Ricci curvature of M is constant, then, we have that either Ric ≡ 1 andM is isometric to an equator or, n is odd,Ric ≡ n−3 n−2 andM is isometric to S n−1 2 ( √ 2 2 )×S n−1 2 ( √ 2 2 ). Next, we will prove that there exists a positive number ̄(n) such that if ...

Immersed Spheres of Finite Total Curvature into Manifolds

We prove that a sequence of, possibly branched, weak immersions of the two-sphere S into an arbitrary compact riemannian manifold (M, h) with uniformly bounded area and uniformly bounded L−norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of S and whose image is made of a connected union of f...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

On the Average of the Scalar Curvature of Minimal Hypersurfaces of Spheres with Low Stability Index

In this paper we show that if the stability index of M is equal to n+2, then the average of the function |A|2 is less than or equal to n − 1. Moreover, if this average is equal to n − 1, then M must be isometric to a Clifford minimal hypersurface.