Monotonicity inequalities for ther-area and a degeneracy theorem forr-minimal graphs

We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-minimal hypersurface in Euclidean space under appropriate quasi-positivity assumptions on certain invariants of the immersion. The proofs are based on the corresponding first variational formula. As an application, we derive a degeneracy theorem for an entire r-minimal graph whose defining function ƒ has first and second derivatives decaying fast enough at infinity: Its Hessian operator D² ƒ has at least n − r null eigenvalues everywhere.     

Uniqueness of H-surfaces in {\mathbb{H}}^2 \times \mathbb{R},{{\vert H\vert \leq 1/2}} , with boundary one or two parallel horizontal circles

We prove that a H-surface M in ${\mathbb{H}}^2 \times {\mathbb{R}} ,\vert H\vert \leq 1/2$ , inherits the symmetries of its boundary $\partial M,$ when $\partial M$ is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to π. Furthermore we prove that the asymptotic boundary of a surface with mean curvature bounded away from zero consists of parts of straight lines, provided it is sufficiently regular.     

On the compactness of constant mean curvature hypersurfaces with finite total curvature

In this work we consider a complete submanifold M with parallel mean curvature vector h immersed in a space form of constant sectional curvature c ￡ 0c\leq 0. If M has finite total curvature and |H|² > -c|H|^2>-c, we prove that M must be compact.     

Hypersurfaces with {H_{r+1}} = 0 in {\mathbb{H}^n \times \mathbb{R}}

We prove the existence of rotational hypersurfaces in \({\mathbb{H}^n \times \mathbb{R}}\) with \({H_{r+1} = 0}\) (r-minimal hupersurfaces) and we classify them. Then we prove some uniqueness theorems for r-minimal hypersurfaces with a given (finite or asymptotic) boundary. In particular, we obtain a Schoen-type theorem for two ended complete hypersurfaces.     

On the Gauss map of hypersurfaces with constant scalar curvature in spheres

In this work we consider connected, complete and orientable hypersurfaces of the sphere with constant nonnegative -mean curvature. We prove that under subsidiary conditions, if the Gauss image of is contained in a closed hemisphere, then is totally umbilic.