semi-Riemannian Warped Product Research Articles

Rigidity and nonexistence of complete hypersurfaces via Liouville type results and other maximum principles, with applications to entire graphs

We investigate complete hypersurfaces with some positive higher order mean curvature in a semi-Riemannian warped product space. Under standard curvature conditions on the ambient space and appropriate constraints on the higher order mean curvatures, we establish rigidity and nonexistence results via Liouville type results and suitable maximum principles related to the divergence of smooth vector fields on a complete noncompact Riemannian manifold. Applications to standard warped product models, like the Schwarzschild, Reissner-Nordström and pseudo-hyperbolic spaces, as well as steady state type spacetimes, are given and a particular study of entire graphs is also presented.

On the rigidity of mean curvature flow solitons in certain semi-Riemannian warped products

We obtain rigidity results concerning complete noncompact solitons of the mean curvature flow related to a nonsingular Killing vector field $K$ globally defined in a semi-Riemannian space, which can be modeled as a warped product whose base corresponds to a fixed integral leaf of the distribution orthogonal to $K$ and the warping function is equal to $|K|$. Our approach is based on a suitable maximum principle dealing with a notion of convergence to zero at infinity. As application, we study the uniqueness of solutions for the mean curvature flow soliton equation in these ambient spaces.

Quasi-Einstein manifolds with structure of warped product

In this paper we prove that, under certain conditions, in a quasi-Einstein semi-Riemannian warped product the fiber is necessarily an Einstein manifold. We provide all the quasi-Einstein manifolds when r-Bakry-Emery tensor is null, the base is conformal to an n-dimensional pseudo-Euclidean space, invariant under the action of an (n−1)-dimensional translation group and the fiber is Ricci-flat. As an application, we have built a family of Ricci-flat Einstein warped product whose base is not locally conformally flat.

A fundamental theorem for submanifolds in semi-Riemannian warped products

In this paper we find necessary and sufficient conditions for a nondegenerate arbitrary signature manifold to be realized as a submanifold in the large class of warped product manifolds. This comprises the case of hypersurfaces immersed in semi-Riemannian warped products proved by M.A. Lawn and M. Ortega in 2015.

Gradient Ricci Solitons with Structure of Warped Product

In this paper we consider semi-Riemannian warped product gradient Ricci solitons. We prove that the potential function depends only on the base and the fiber is necessarily Einstein manifold. We provide all such solutions in the case of steady gradient Ricci solitons when the base is conformal to an n-dimensional pseudo-Euclidean space, invariant under the action of an (n − 1)-dimensional translation group, and the fiber is Ricci-flat.

Rigidity of complete hypersurfaces in warped product spaces via higher order mean curvatures

In this article, we deal with complete hypersurfaces immersed in a semi-Riemannian warped product space, which is supposed to obey standard convergence conditions. Under appropriated constraints on the higher order mean curvatures, but without the requirement of the existence of an elliptic point, we are able to obtain sufficient conditions which guarantee that such a hypersurface must be a slice of the ambient space. Our approach is based on a suitable generalized maximum principle due to Alias et al. (Math Proc Camb Philos Soc 152:365–383, 2012) applied to the linearized operators of the higher order mean curvatures.

Complete spacelike hypersurfaces with positive r-th mean curvature in a semi-Riemannian warped product

Abstract In this paper, by supposing a natural comparison inequality on the positive r-th mean curvatures of the hypersurface, we obtain some new Bernstein-type theorems for complete spacelike hypersurfaces immersed in a semi-Riemannian warped product of constant sectional curvature. Generalizing the above results, under a restriction on the sectional curvature or the Ricci curvature tensor of the fiber of a warped product, we also prove some new rigidity theorems in semi-Riemannian warped products. Our main results extend some recent Bernstein-type theorems proved in [12, 13, 14].

A fundamental theorem for hypersurfaces in semi-Riemannian warped products

We give necessary and sufficient conditions for a semi-Riemannian manifold of arbitrary signature to be locally isometrically immersed into certain warped products. Then, we describe a way to use the structure equations of such immersions to construct foliations of marginally trapped surfaces in a four-dimensional Lorentzian spacetimes. We point out that, sometimes, Gauss and Codazzi equations are not sufficient to ensure the existence of a local isometric immersion of a semi-Riemannian manifold as a hypersurface of another manifold. We finally give two low-dimensional examples to illustrate our results.

On Complete Spacelike Hypersurfaces in a Semi-Riemannian Warped Product

By applying Omori-Yau maximal principal theory and supposing an appropriate restriction on the norm of gradient of height function, we obtain some new Bernstein-type theorems for complete spacelike hypersurfaces with nonpositive constant mean curvature immersed in a semi-Riemannian warped product. Furthermore, some applications of our main theorems for entire vertical graphs in Robertson-Walker spacetime and for hypersurfaces in hyperbolic space are given.

On Bernstein-Type Theorems in Semi-Riemannian Warped Products

Complete spacelike hypersurfaces immersed in semi-Riemannian warped products are investigated. By using a technique according to Yau (1976) and a reasonable restriction on the mean curvature of the hypersurfaces, we obtain some new Bernstein-type theorems which extend some known results proved by Camargo et al. (2011) and Colares and Lima (2012).

On the Unicity of Complete Hypersurfaces Immersed in a Semi-Riemannian Warped Product

Our purpose is to apply appropriate generalized maximum principles in order to study the unicity of complete hypersurfaces immersed in a semi-Riemannian warped product, which is supposed to obey a suitable convergence condition. In this setting, by assuming a natural comparison inequality between the r-th mean curvatures of the hypersurface and the ones of the slices of the slab where the hypersurface is contained, we establish sufficient conditions to guarantee that such a hypersurface must be a slice.

On the mean curvature of semi-Riemannian graphs in semi-Riemannian warped products

We investigate the mean curvature of semi-Riemannian graphs in the semi-Riemannian warped product M×fℝe , where M is a semi-Riemannian manifold, ℝe is the real line ℝ with metric edt2 (e = ±1), and f: M→ℝ + is the warping function. We obtain an integral formula for mean curvature and some results dealing with estimates of mean curvature, among these results is a Heinz–Chern type inequality.

Some rigidity theorems in semi-Riemannian warped products

We study the problem of uniqueness of complete hypersurfaces immersed in a semi-Riemannian warped product whose warping function has convex logarithm. By applying a maximum principle at the infinity due to S. T. Yau and supposing a natural comparison inequality between the mean curvature of the hypersurface and that of the slices of the region where the hypersurface is contained, we obtain rigidity theorems in such ambient spaces. Applications to the hyperbolic and the steady state spaces are given.

Complete hypersurfaces immersed in a semi-Riemannian warped product

The aim of this paper is to study the uniqueness of complete hypersurfaces immersed in a semi-Riemannian warped product whose warping function has convex logarithm and such that its fiber has constant sectional curvature. By using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds and supposing a natural comparison inequality between the r-th mean curvatures of the hypersurface and that ones of the slices of the region where the hypersurface is contained, we are able to prove that a such hypersurface must be, in fact, a slice.

Bernstein-type theorems in semi-Riemannian warped products

This paper deals with complete hypersurfaces immersed in the $(n+1)$-dimensional hyperbolic and steady state spaces. By applying a technique of S. T. Yau and imposing suitable conditions on both the $r$-th mean curvatures and on the norm of the gradient of the height function, we obtain Bernstein-type results in each of these ambient spaces.

A correspondence for isometric immersions into product spaces and its applications

We present a correspondence for isometric immersions that are graphs in Riemannian or semi-Riemannian warped product spaces. We use this correspondence to give several existence and non-existence theorems for hypersurfaces in Riemannian or Lorentzian spaces. In the case of surfaces, we obtain further applications regarding height estimates, harmonic representation of surfaces and the existence of holomorphic quadratic differentials in homogeneous and non-homogeneous spaces.

Complete Vertical Graphs with Constant Mean Curvature in Semi-Riemannian Warped Products

In this paper we study complete vertical graphs of constant mean curvature in the Hyperbolic and Steady State spaces. We first derive suitable formulas for the Laplacians of the height function and of a support-like function naturally attached to the graph; then, under appropriate restrictions on the values of the mean curvature and the growth of the height function, we obtain necessary conditions for the existence of such a graph. In the two-dimensional case we apply this analytical framework to state and prove Bernstein-type results in each of these ambient spaces. This paper deals with complete non-compact constant mean curvature graphs over a horosphere of the Hyperbolic space, as well as over horizontal hyperplanes (slices) in the Steady State space. In connection with our work, L. Al´oas and M. Dajczer (cf. (2)) studied properly immersed complete surfaces of the 3−dimensional Hyperbolic space contained between two horospheres, obtaining a Bernstein-type result for the case of constant mean curvature between −1 and 1. In de Sitter space, K. Akutagawa (cf. (5)) proved that complete spacelike hypersurfaces having constant mean curvature in a specific interval of the real line are totally umbilical. Also for de Sitter space, among other interesting results S. Montiel (cf. (14)) proves that, under an appropriate restriction on their Hyperbolic Gauss map, complete spacelike Hypersurfaces of constant mean curvature greater than or equal to 1 must actually have mean curvature 1. For the Lorentz case, our motivation to restrict attention to the Steady State space comes from the fact that there exists a natural duality between the Gauss maps of Riemannian hypersurfaces of this space and those of the Hyperbolic space, provided we model these as hyperquadrics of the Lorentz-Minkowski space (cf. section 5). Besides, in physical context the Steady State space appears naturally as an exact solution for the Einstein equations, being a cosmological model where matter is supposed to travel along geodesics normal to horizontal hyperplanes; these, in turn, serve as the initial data for the Cauchy problem associated to those equations (cf. (8), chapter 5).