arXiv subjects
Publications and source records attributed to Maria Fernanda Elbert.
In this paper we use the notion of stability for free boundary surfaces with constant higher order mean curvature to obtain rigidity results for $H_2$-surfaces with free boundary of a geodesic ball of a simply connected $3$-dimensional space form or a slab of $\mathbb{R}^3$.
We propose a notion of stability for capillary hypersurfaces with constant higher order mean curvature and we generalize some results of the classical stability theory for CMC capillary hypersurfaces.
We propose a notion of stability for constant k-mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem. Our approach led us to work with two different stability operators and we are able to relate stability to the study of the respective first eigenvalues. Moreover, we prove that embedded rotational spheres with constant k-mean curvature in Hnx R or in SnxR are not stable.
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.
We prove that strong finite total curvature complete hypersurfaces of (n+1)-euclidean space are proper and diffeomorphic to a compact manifold minus finitely many points. With an additional condition, we also prove that the Gauss map of such hypersurfaces extends continuously to the punctures.
We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with $H_{r+1}=0$ 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.
In this paper, we are concerned with hypersurfaces in $H^n\times R$ with constant r-mean curvature, to be called $H_r$-hypersurfaces. We construct examples of complete $H_r$-hypersurfaces which are invariant by parabolic screw motion or by rotation. We prove that there is a unique rotational strictly convex entire $H_r$-graph for each value $0 \frac{n-r}{n}$, there is a unique embedded compact strictly convex rotational $H_r$-hypersurface. By using them as barriers, we obtain some interesting geometric results, including height estimates and an Alexandrov-type Theorem. Namely, we prove that an embedded compact $H_r$-hypersurface in $H^n\times R$ is rotational ($H_r>0$).