On the structure of hypersurfaces in Hn×R\mathbb{H}^n\times \mathbb{R} with finite strong total curvature

We prove that if $X:M^n\to\mathbb{H}^n\times \mathbb{R}$, $n\geq 3$, is a an orientable, complete immersion with finite strong total curvature, then $X$ is proper and $M$ is diffeomorphic to a compact manifold $\bar M$ minus a finite number of points $q_1, \dots q_k$. Adding some extra hypothesis, including $H_r=0,$ where $H_r$ is a higher order mean curvature, we obtain more information about the geometry of a neighbourhood of each puncture. The reader will also find in this paper a classification result for the hypersurfaces of $\mathbb{H}^n\times \mathbb{R}$ which satisfy $H_r=0$ and are invariant by hyperbolic translations and a maximum principle in a half space for these hypersurfaces

Caccioppoli's inequalities on constant mean curvature hypersurfaces in Riemannian manifolds

This is a revised version (minor changes and a deeper insight in the positive curvature case). We prove some Caccioppoli's inequalities for the traceless part of the second fundamental form of a complete, noncompact, finite index, constant mean curvature hypersurface of a Riemannian manifold, satisfying some curvature conditions. This allows us to unify and clarify many results scattered in the literature and to obtain some new results. For example, we prove that there is no stable, complete, noncompact hypersurface in ${\mathbb R}^{n+1},$ $n\leq 5,$ with constant mean curvature $H\not=0,$ provided that, for suitable $p,$ the $L^p$-norm of the traceless part of second fundamental form satisfies some growth condition.Comment: 31 page

A note on the stability for constant higher mean curvature hypersurfaces in a Riemannian manifold

We give a notion of stability for constant r-mean curvature hypersurfaces in a general Riemannian manifold. When the ambient manifold is a space form, our notion coincide with the variational one \cite{BC} and when r=1, it coincides with the classic one for constant mean curvature hypersurfaces.Comment: We added a section where we introduce a symmetrized r-stability operato