Complete Stable Hypersurface Research Articles

Stable hypersurfaces with zero scalar curvature in Euclidean space

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the radius of the smallest extrinsic ball which contains the domain.

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

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. Keywords: stable hypersurface, constant mean curvature, isometric immersion, Bernstein theorem.

$L^2$ harmonic forms on a complete stable hypersurfaces with constant mean curvature

We show that an ^-dimensional (2 < n < 5) complete noncompact strongly stable hypersurface M with constant mean curvature in an (n + 1)-dimensional manifold M of nonnegative bi-Ricci curvature admits no nontrivial L harmonic 1-forms.