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Alma L. Albujer

Publications and source records attributed to Alma L. Albujer.

16 recordsLinked to original sources
arXiv2024

Non-degenerate anisocurved surfaces in homogeneous 3-manifolds

In this manuscript we consider non-degenerate surfaces $Σ^2$ immersed in a 3-dimensional homogeneous space $\mathbb{L}^3(κ,τ)$ endowed with two different metrics, the one induced by the Riemannian metric of $\mathbb{E}^3(κ,τ)$ and the non-degenerate metric inherited by the Lorentzian one of $\mathbb{L}^3(κ,τ)$. Therefore, we have two different geometries on $Σ^2$ and we can compare them. In particular, we can consider the Gaussian curvature functions which respect to both metrics and study the geometry of the surfaces satisfying that both Gaussian curvature functions are opposite. We will call these surfaces anisocurved surfaces. In order to obtain our main results we also need to impose some extra assumptions regarding the extrinsic curvatures with respect to both metrics.

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arXiv2024

Willmore surfaces and Hopf tori in homogeneous 3-manifolds

Some classification results for closed surfaces in Berger spheres are presented. On the one hand, a Willmore functional for isometrically immersed surfaces into an homogeneous space $\mathbb{E}^{3}(κ,τ)$ with isometry group of dimension $4$ is defined and its first variational formula is computed. Then, we characterize Clifford and Hopf tori as the only Willmore surfaces satifying a sharp Simons-type integral inequality. On the other hand, we also obtain some integral inequalities for closed surfaces with constant extrinsic curvature in $\mathbb{E}^3(κ,τ)$, becoming equalities if and only if the surface is a Hopf torus in a Berger sphere.

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arXiv2024

A Moser-Bernstein problem for Riemannian warped products

In this work we deal with an elliptic non-linear problem, which arises naturally from Riemannian geometry. This problem has clasically been studied in the the Euclidean $n$-dimensional space and it is known as the Moser-Bernstein problem. Nevertheless we solve this type of problems in a wide family of Riemannian manifolds, constructed as Riemannian warped products. More precicely, we study the entire solutions to the minimal hypersurface equation in a Riemannian warped product $M=P\times_h\mathbb{R}$, where $P$ is a complete Riemannian parabolic manifold and $h$ a positive smooth function on $P$.

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arXiv2024

Total mean curvature surfaces in the product space $\mathbb{S}^n\times\mathbb{R}$ and applications

The total mean curvature functional for submanifolds into the Riemannian product space $\mathbb{S}^n\times\mathbb{R}$ is considered and its first variational formula is presented. Later on, two second order differential operators are defined and a nice integral inequality relating both of them is proved. Finally we prove our main result: an integral inequality for closed stationary $\mathcal{H}$-surfaces in $\mathbb{S}^n\times\mathbb{R}$, characterizing the cases where the equality is attained.

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arXiv2024

On complete trapped submanifolds in globally hyperbolic spacetimes

The aim of this manuscript is to obtain rigidity and non-existence results for parabolic spacelike submanifolds with causal mean curvature vector field in orthogonally splitted spacetimes, and in particular, in globally hyperbolic spacetimes. We also obtain results regarding the geometry of submanifolds by ensuring, under some mild hypothesis, the non-existence of local minima or maxima of certain distinguished function. Furthermore, in this last case the submanifold does not need to be parabolic or even complete. As an application in General Relativity, we obtain several nice results regarding (non-necessarily closed) trapped surfaces in a huge family of spacetimes. In fact, we show how our technique allows us to recover some relevant previous results for trapped surfaces in both, standard static spacetimes and Generalized Robertson-Walker spacetimes.

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arXiv2022

Critical points of the solution to the $H_R=H_L$ surface equation

Spacelike surfaces with the same mean curvature in $\mathbb{R}^3$ and $\mathbb{L}^3$ are locally described as the graph of the solutions to the $H_R=H_L$ surface equation, which is an elliptic partial differential equation except at the points at which the gradient vanishes, because the equation degenerates. In this paper we study precisely the critical points of the solutions to such equation. Specifically, we give a necessary geometrical condition for a point to be critical, we obtain a new uniqueness result for the Dirichlet problem related to the $H_R=H_L$ surface equation and we get a Heinz-type bound for the inradius of the domain of any solution to such equation, improving a previous result by the authors. Finally, we also get a bound for the inradius of the domain of any function of class $\mathcal{C}^2$ in terms of the curvature of its level curves.

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arXiv2022

On the symmetries of a Kaehler manifold

In this manuscript we study natural symmetries of Kaehler manifolds: constant holomorphic sectional curvature Kaheler manifolds, semisymmetric Kaehler manifolds and holomorphically pseudosymmetric Kaehler manifolds. We get characterization results, as well as a geometric interpretation of the complex Tachibana tensor.

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arXiv2016

Rigidity of spacelike hypersurfaces in spatially weighted generalized Robertson-Walker spacetimes

Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weight function and on the f-mean curvature, we establish sufficient conditions to guarantee that such a hypersurface must be a slice of the ambient space. In this setting, we also obtain new Calabi-Bernstein type results concerning entire graphs in a spatially weighted GRW spacetime.

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arXiv2016

Geometric properties of surfaces with the same mean curvature in R^3 and L^3

Spacelike surfaces in the Lorentz-Minkowski space L^3 can be endowed with two different Riemannian metrics, the metric inherited from L^3 and the one induced by the Euclidean metric of R^3. It is well known that the only surfaces with zero mean curvature with respect to both metrics are open pieces of the helicoid and of spacelike planes. We consider the general case of spacelike surfaces with the same mean curvature with respect to both metrics. One of our main results states that those surfaces have non-positive Gaussian curvature in R^3. As an application of this result, jointly with a general argument on the existence of elliptic points, we present several geometric consequences for the surfaces we are considering. Finally, as any spacelike surface in L^3 is locally a graph over a domain of the plane x_3=0, our surfaces are locally determined by the solutions to the H_R=H_L surface equation. Some uniqueness results for the Dirichlet problem associated to this equation are given.

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arXiv2016

$ϕ$-parabolicity and the uniqueness of spacelike hypersurfaces immersed in a spatially weighted GRW spacetime

In this paper, we extend a technique due to Romero, Rubio and Salamanca establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed in a weighted generalized Robertson-Walker spacetime whose fiber has phi-parabolic universal Riemannian covering. As some applications of this criteria, we obtain uniqueness results concerning spacelikes hypersurfaces immersed in spatially weighted generalized Robertson-Walker spacetimes. Furthermore, Calabi-Bernstein type results are also given.

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arXiv2009

Parabolicity of maximal surfaces in Lorentzian product spaces

In this paper we establish some parabolicity criteria 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 with non-negative Gaussian curvature and $M^2\times\mathbb{R}_1$ is endowed with the Lorentzian product metric $<,>=<,>_M-dt^2$. In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain $Ω\subseteq M$ is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi-Bernstein result for entire maximal graphs in $M^2\times\mathbb{R}_1$.

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arXiv2009

On the scalar curvature of hypersurfaces in spaces with a Killing field

We consider compact hypersurfaces in an $(n+1)$-dimensional either Riemannian or Lorentzian space $N^{n+1}$ endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the simpler case when $N=M^n\times R$ is a product space, allows us to derive some interesting consequences in terms of the scalar curvature of the hypersurface. For instance, when $n=2$ and $M^2$ is either the sphere $\mathbb{S}^2$ or the real projective plane $\mathbb{RP}^2$, we characterize the slices of the trivial totally geodesic foliation $M^2\times\{t\}$ as the only compact two-sided surfaces with constant Gaussian curvature in the Riemannian product $M^2\times\mathbb{R}$ such that its angle function does not change sign. When $n\geq 3$ and $M^n$ is a compact Einstein Riemannian manifold with positive scalar curvature, we also characterize the slices as the only compact two-sided hypersurfaces with constant scalar curvature in the Riemannian product $M^n\times\mathbb{R}$ whose angle function does not change sign. Similar results are also established for spacelike hypersurfaces in a Lorentzian product $\mathbb{M}\times\mathbb{R}_1$.

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arXiv2009

A local estimate for maximal surfaces in Lorentzian product spaces

In this paper we introduce a local approach for the study of maximal surfaces immersed into a Lorentzian product space of the form $M^2\times R_1$, where $M^2$ is a connected Riemannian surface and $M^2\times R_1$ is endowed with the product Lorentzian metric. Specifically, we establish a local integral inequality for the squared norm of the second fundamental form of the surface, which allows us to derive an alternative proof of our Calabi-Bernstein theorem given in \cite{AA}.

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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.

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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.

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