arXiv subjects

Luis J. Alias

Publications and source records attributed to Luis J. Alias.

24 records · Page 2Linked to original sources

arXiv2009

Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces

In this paper we establish new Calabi-Bernstein results for maximal surfaces immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$, where $M^2$ is a connected Riemannian surface and $M^2\times\mathbb{R}_1$ is endowed with the Lorentzian metric $<,>=<,>_{M}-dt^2$. In particular, when $M$ is a Riemannian surface with non-negative Gaussian curvature $K_M$, we prove that any complete maximal surface in $M^2\times\mathbb{R}_1$ must be totally geodesic. Besides, if $M$ is non-flat we conclude that it must be a slice $M\times\{t_0\}$, $t_0\in\mathbb{R}$ (here by "complete" it is meant, as usual, that the induced Riemannian metric on the maximal surface from the ambient Lorentzian metric is complete). We prove that the same happens if the maximal surface is complete with respect to the metric induced from the Riemannian product $M^2\times\mathbb{R}$. This allows us to give also a non-parametric version of the Calabi-Bernstein theorem for entire maximal graphs in $M^2\times\mathbb{R}_1$, under the same assumptions on $K_M$. Moreover, we also construct counterexamples which show that our Calabi-Bernstein results are no longer true without the hypothesis $K_M\geq 0$. These examples are constructed via a duality result between minimal and maximal graphs.

math.DG↗

arXiv2008

A characterization of quadric constant mean curvature hypersurfaces of spheres

Let $ϕ:M\to\mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be an immersion of a complete $n$-dimensional oriented manifold. For any $v\in\mathbb{R}^{n+2}$, let us denote by $\ell_v:M\to\mathbb{R}$ the function given by $\ell_v(x)=ϕ(x),v$ and by $f_v:M\to\mathbb{R}$, the function given by $f_v(x)=ν(x),v$, where $ν:M\to\mathbb{S}^{n}$ is a Gauss map. We will prove that if $M$ has constant mean curvature, and, for some $v\ne{\bf 0}$ and some real number $λ$, we have that $\ell_v=λf_v$, then, $ϕ(M)$ is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface $M^n$ in $\mathbb{S}^{n+1}$ which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to $2n+4$.

math.DG↗

arXiv2008

Spacelike hypersurfaces with constant mean curvature in the steady state space

We consider complete spacelike hypersurfaces with constant mean curvature in the open region of de Sitter space known as the steady state space. We prove that if the hypersurface is bounded away from the infinity of the ambient space, then the mean curvature must be H=1. Moreover, in the 2-dimensional case we obtain that the only complete spacelike surfaces with constant mean curvature which are bounded away from the infinity are the totally umbilical flat surfaces. We also derive some other consequences for hypersurfaces which are bounded away from the future infinity. Finally, using an isometrically equivalent model for the steady state space, we extend our results to a wider family of spacetimes.

math.DG↗

arXiv2007

Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes

In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker (GRW) spacetimes. In particular, we consider the following question: Under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, esentially, under the so called convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.

math.DG↗

arXiv2006

The Dirichlet problem for constant mean curvature surfaces in Heisenberg space

We study constant mean curvature graphs in the Riemannian 3-dimensional Heisenberg spaces ${\cal H}={\cal H}(τ)$. Each such ${\cal H}$ is the total space of a Riemannian submersion onto the Euclidean plane $\mathbb{R}^2$ with geodesic fibers the orbits of a Killing field. We prove the existence and uniqueness of CMC graphs in ${\cal H}$ with respect to the Riemannian submersion over certain domains $Ω\subset\mathbb{R}^2$ taking on prescribed boundary values.

math.DG↗

arXiv2003

Constant higher order mean curvature hypersurfaces in Riemannian spaces

It is still an open question whether a compact embedded hypersurface in the Euclidean space R^{n+1} with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of surfaces in R^3. In a recent paper the first and third authors have shown that this is true for the case of hypersurfaces in R^{n+1} with constant scalar curvature, and more generally, hypersurfaces with constant higher order r-mean curvature, when r>1. In this paper we deal with some aspects of the classical problem above, by considering it in a more general context. Specifically, our starting general ambient space is an orientable Riemannian manifold, where we will consider a general geometric configuration consisting of an immersed hypersurface with boundary on an oriented hypersurface P. For such a geometric configuration, we study the relationship between the geometry of the hypersurface along its boundary and the geometry of its boundary as a hypersurface of P, as well as the geometry of P. Our approach allows us to derive, among others, interesting results for the case where the ambient space has constant curvature. In particular, we are able to extend the previous symmetry results to the case of hypersurfaces with constant higher order r-mean curvature in the hyperbolic space and in the sphere.

math.DG↗