Calabi Bernstein Type Research Articles

Spacelike translating solitons of mean curvature flow with forcing term in Lorentzian product spaces: Nonexistence, mean convexity, rigidity and Calabi-Bernstein type results

Spacelike translating solitons of mean curvature flow with forcing term in Lorentzian product spaces: Nonexistence, mean convexity, rigidity and Calabi-Bernstein type results

Calabi-Bernstein type results for critical points of a weighted area functional in [formula omitted] and [formula omitted]

In this paper we prove some Calabi-Bernstein type and non-existence results concerning complete [φ,e→3]-minimal surfaces in R3 whose Gauss maps lie on compacts subsets of open hemispheres of S2. We also give a general non-existence result for complete spacelike [φ,e→3]-maximal surfaces in L3 and, in particular, we obtain a Calabi-Bernstein type result when φ˙ is bounded.

Multiplicity and Calabi–Bernstein type asymptotic property of positive solutions for one-dimensional Minkowski-curvature problems

Multiplicity and Calabi–Bernstein type asymptotic property of positive solutions for one-dimensional Minkowski-curvature problems

Bifurcation and Calabi-Bernstein type asymptotic property of solutions for the one-dimensional Minkowski-curvature equation

In this paper, we investigate the global structure of bifurcation branches of one-sign solutions and sign-changing solutions for one-dimensional Minkowski-curvature problems with a strong singular weight. Our interest of the nonlinearity is either linear or sublinear near zero. Growth conditions near ∞ are not necessary and the proofs are mainly employed by bifurcation theories based on Whyburn's limit argument and analysis techniques. We also show Calabi-Bernstein type asymptotic property of one-sign solutions by proving that one-sign solutions on two bifurcation branches converge to two linear functions.

Spacelike translating solitons in Lorentzian product spaces: Nonexistence, Calabi–Bernstein type results and examples

We establish nonexistence results for complete spacelike translating solitons immersed in a Lorentzian product space [Formula: see text], under suitable curvature constraints on the curvatures of the Riemannian base [Formula: see text]. In particular, we obtain Calabi–Bernstein type results for entire translating graphs constructed over [Formula: see text]. For this, we prove a version of the Omori–Yau’s maximum principle for complete spacelike translating solitons. Besides, we also use other two analytical tools related to an appropriate drift Laplacian: a parabolicity criterion and certain integrability properties. Furthermore, under the assumption that the base [Formula: see text] is non-positively curved, we close our paper constructing new examples of rotationally symmetric spacelike translating solitons embedded into [Formula: see text].

Uniqueness and nonexistence of complete spacelike hypersurfaces, Calabi–Bernstein type results and applications to Einstein–de Sitter and steady state type spacetimes

We investigate the geometry of complete spacelike hypersurfaces (immersed) in a generalized Robertson-Walker spacetime $$-I\times _fM^{n}$$ . Under suitable constraints on the sectional curvature of the Riemannian fiber $$M^n$$ , on the warping function f and on the future mean curvature (that is, the mean curvature function with respect to the future-pointing Gauss map of the spacelike hypersurface), we are able to prove that such a spacelike hypersurface must be a slice $$\{t\}\times M^{n}$$ of the ambient spacetime. Nonexistence and Calabi–Bernstein type results concerning entire spacelike graphs constructed over the Riemannian fiber $$M^n$$ are also obtained, as well as applications to the Einstein–de Sitter and steady state type spacetimes are given. Our approach is based on the so-called Omori–Yau’s generalized maximum principle and on certain integrability properties due to Yau.

Rigidity of parabolic and stable spacelike hypersurfaces in a generalized Robertson–Walker spacetime

In this paper, we deal with a parabolic stable spacelike hypersurface immersed in a generalized Robertson–Walker spacetime. We consider the support type function as the product between the warping function and the hyperbolic cosine of the hyperbolic angle of the normal vector field with the timelike parallel direction $$\partial _t$$ . We provide some conditions under which the support type function must be constant or, particularly, when the hypersurface must be a slice, yielding a Calabi–Bernstein type result.

New Calabi–Bernstein type results in Lorentzian product spaces with density

We investigate the rigidity of complete spacelike hypersurfaces immersed in a weighted Lorentzian product space of the type R1×Pfn, endowed with a weight function f which does not depend on the parameter t∈R and whose Riemannian fiber Pn has nonnegative Bakry–Émery–Ricci tensor. In this direction, supposing that the f-mean curvature is constant and assuming appropriate constraints on the norm of the gradient of f, we prove that such a spacelike hypersurface must be a slice of the ambient space. As application of our investigation, we obtain new Calabi–Bernstein type results concerning entire spacelike graphs constructed over Pn. Our approach is based on an extension for the drift Laplacian of a generalized maximum principle of Akutagawa (1987) and a Liouville-type result due to Pigola et al. (2005).

Calabi–Bernstein type results for complete constant mean curvature spacelike hypersurfaces in spatially open Generalized Robertson–Walker spacetimes

In this article we study complete constant mean curvature spacelike hypersurfaces in spatially open Generalized Robertson–Walker spacetimes. Under appropriate geometric assumptions we prove new parametric uniqueness results as well as obtain very general Calabi–Bernstein type results for the constant mean curvature equation for spacelike graphs in these ambient spacetimes.

An Extension of Calabi’s Correspondence between the Solutions of Two Bernstein Problems to More General Elliptic Nonlinear Equations

A new correspondence between the solutions of theminimal surface equation in a certain 3-dimensional Riemannian warped product and the solutions of the maximal surface equation in a 3-dimensional standard static space-time is given. This widely extends the classical duality between minimal graphs in 3-dimensional Euclidean space and maximal graphs in 3-dimensional Lorentz–Minkowski space-time. We highlight the fact that this correspondence can be restricted to the respective classes of entire solutions. As an application, a Calabi–Bernstein-type result for certain static standard space-times is proved.

$$\phi $$ 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 et al. (Class Quantum Gravity 30:1–13, 2013; Int J Geom Methods Mod Phys 10:1360014, 2013; J Math Anal Appl 419:355–372, 2014) 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 spacelike hypersurfaces immersed in spatially weighted generalized Robertson–Walker spacetimes. Furthermore, Calabi–Bernstein type results are also given.

A New Calabi–Bernstein Type Result in Spatially Closed Generalized Robertson–Walker Spacetimes

The aim of this article is to study the uniqueness of a complete spacelike hypersurface $${\sum^{n}}$$ immersed with constant mean curvature H in a spatially closed generalized Robertson–Walker spacetime $${\overline{M}^{n+1} = -I {\times_{f}} {M^{n}}}$$ , whose Riemannian fiber $${M^n}$$ has positive curvature. Supposing that the warping function f is such that −log f is convex and $${H{f^{\prime}} \leq 0}$$ along $${\sum^{n}}$$ , we show that $${\sum^{n}}$$ must be isometric to a totally geodesic slice of $${\overline{M}^{n+1}}$$ . When $${\overline{M}^{n+1}}$$ is a Lorentzian product space, we obtain a new Calabi–Bernstein type result concerning the CMC spacelike hypersurface equation.

On entire f-maximal graphs in the Lorentzian product Gn×R1

In the Lorentzian product G n × R 1 , we give a comparison theorem between the f -volume of an entire f -maximal graph and the f -volume of the hyperbolic H r + under the condition that the gradient of the function defining the graph is bounded away from 1. This condition comes from an example of non-planar entire f -maximal graph in G n × R 1 and is equivalent to the hyperbolic angle function of the graph being bounded. As a consequence, we obtain a Calabi–Bernstein type theorem for f -maximal graphs in G n × R 1 .

Rigidity of complete 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 spacetime. In this setting, we also obtain new Calabi–Bernstein type results concerning entire graphs in a spatially weighted GRW spacetime.

Complete spacelike hypersurfaces in generalized Robertson–Walker and the null convergence condition: Calabi–Bernstein problems

We study constant mean curvature spacelike hypersurfaces in generalized Robertson–Walker spacetimes $$\overline{M}= I \times _f F$$ which are spatially parabolic (i.e. its fiber F is a (non-compact) complete Riemannian parabolic manifold) and satisfy the null convergence condition. In particular, we provide several rigidity results under appropriate mathematical and physical assumptions. We pay special attention to the case where the GRW spacetime is Einstein. As an application, some Calabi–Bernstein type results are given.

New Calabi–Bernstein type results in weighted generalized Robertson–Walker spacetimes

We apply suitable generalized maximum principles in order to obtain new Calabi–Bernstein’s type results concerning complete spacelike hypersurfaces immersed in a weighted generalized Robertson–Walker spacetime. Assuming a natural comparison inequality between the weighted mean curvatures of the hypersurface and those of the slices of the timelike bounded region where the hypersurface is supposed to be contained, we get sufficient conditions which guarantee that such a hypersurface must be a slice. Furthermore, we also treat the case when the ambient spacetime is static.

Constant mean curvature spacelike hypersurfaces in Lorentzian warped products and Calabi–Bernstein type problems

In this paper we provide several uniqueness and non-existence results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian warped products under appropriate geometric assumptions. As a consequence of this parametric study, we obtain very general uniqueness and non-existence results for a large family of uniformly elliptic EDP’s, so solving the Calabi–Bernstein problem in a wide family of spacetimes.

A Nonlinear Inequality Arising in Geometry and Calabi-Bernstein Type Problems

A characterization for the entire solutions of a nonlinear inequality, which has a natural interpretation in terms of certain nonflat Robertson-Walker spacetimes, is given. As an application, new Calabi-Bernstein type problems are solved.

Uniqueness of maximal surfaces in Generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems

Complete maximal surfaces in Generalized Robertson–Walker spacetimes obeying either the Null Convergence Condition or the Timelike Convergence Condition are studied. Uniqueness theorems that widely extend the classical Calabi–Bernstein theorem, as well as previous results on complete maximal surfaces in Robertson–Walker spacetimes, i.e. the case in which the Gauss curvature of the fiber is a constant, are given. All the entire solutions to the maximal surface differential equation in certain Generalized Robertson–Walker spacetimes are found.

Spacelike hypersurfaces of constant mean curvature and Calabi-Bernstein type problems

Spacelike hypersurfaces of constant mean curvature and Calabi-Bernstein type problems