scholarly journals Space-like hypersurfaces with functionally bounded mean curvature in Lorentzian warped products and generalized Calabi–Bernstein-type problems

2018 ◽  
Vol 149 (04) ◽  
pp. 849-868
Author(s):  
Juan A. Aledo ◽  
Rafael M. Rubio ◽  
Juan J. Salamanca
Keyword(s):  
Riemannian Manifold ◽  
Mean Curvature ◽  
Universal Covering ◽  
Complete Riemannian Manifold ◽  
Warped Products ◽  
Bernstein Type ◽  
Rigidity Results

We study space-like hypersurfaces with functionally bounded mean curvature in Lorentzian warped products , where F is a (non-compact) complete Riemannian manifold whose universal covering is parabolic. In particular, we provide several rigidity results under appropriate mathematical and physical assumptions. As an application, several Calabi–Bernstein-type results are obtained which widely extend the previous ones in this setting.

scholarly journals Space-like hypersurfaces with functionally bounded mean curvature in Lorentzian warped products and generalized Calabi–Bernstein-type problems

2018 ◽  
pp. 1-20
Author(s):  
Juan A. Aledo ◽  
Rafael M. Rubio ◽  
Juan J. Salamanca
Keyword(s):  
Riemannian Manifold ◽  
Mean Curvature ◽  
Universal Covering ◽  
Complete Riemannian Manifold ◽  
Warped Products ◽  
Bernstein Type ◽  
Rigidity Results

We study space-like hypersurfaces with functionally bounded mean curvature in Lorentzian warped products , where F is a (non-compact) complete Riemannian manifold whose universal covering is parabolic. In particular, we provide several rigidity results under appropriate mathematical and physical assumptions. As an application, several Calabi–Bernstein-type results are obtained which widely extend the previous ones in this setting.

scholarly journals CONSTANT MEAN CURVATURE HYPERSURFACES IN WARPED PRODUCT SPACES

2007 ◽  
Vol 50 (3) ◽  
pp. 511-526 ◽  
Author(s):  
Luis J. Alías ◽  
Marcos Dajczer
Keyword(s):  
Mean Curvature ◽  
Warped Product ◽  
Constant Mean Curvature ◽  
Complete Riemannian Manifold ◽  
General Situation ◽  
Warped Products ◽  
General Version ◽  
Product Spaces ◽  
Constant Mean Curvature Hypersurfaces ◽  
Warped Product Spaces

AbstractWe study hypersurfaces of constant mean curvature immersed into warped product spaces of the form $\mathbb{R}\times_\varrho\mathbb{P}^n$, where $\mathbb{P}^n$ is a complete Riemannian manifold. In particular, our study includes that of constant mean curvature hypersurfaces in product ambient spaces, which have recently been extensively studied. It also includes constant mean curvature hypersurfaces in the so-called pseudo-hyperbolic spaces. If the hypersurface is compact, we show that the immersion must be a leaf of the trivial totally umbilical foliation $t\in\mathbb{R}\mapsto\{t\}\times\mathbb{P}^n$, generalizing previous results by Montiel. We also extend a result of Guan and Spruck from hyperbolic ambient space to the general situation of warped products. This extension allows us to give a slightly more general version of a result by Montiel and to derive height estimates for compact constant mean curvature hypersurfaces with boundary in a leaf.

A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions

2019 ◽  
Vol 19 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Sergey Stepanov ◽  
Irina Tsyganok
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Type Theorem ◽  
Complete Riemannian Manifold ◽  
Warped Products ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Totally Umbilical

Abstract We prove a Liouville-type theorem for two orthogonal complementary totally umbilical distributions on a complete Riemannian manifold with non-positive mixed scalar curvature. This is applied to some special types of complete doubly twisted and warped products of Riemannian manifolds.

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

2015 ◽  
Vol 23 (2) ◽  
pp. 259-277
Author(s):  
Yaning Wang ◽  
Ximin Liu
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Curvature Tensor ◽  
Mean Curvature ◽  
Warped Product ◽  
Warped Products ◽  
Bernstein Type ◽  
Complete Spacelike Hypersurfaces ◽  
Spacelike Hypersurfaces ◽  
Comparison Inequality

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].

scholarly journals New Bernstein Type Results in Weighted Warped Products

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Ning Zhang
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Warped Products ◽  
Bernstein Type ◽  
Weighted Mean ◽  
Curvature Equation ◽  
Uniqueness Results ◽  
Mean Curvature Equation ◽  
Weighted Mean Curvature ◽  
Entire Graphs

In this paper, we obtain new parametric uniqueness results for complete constant weighted mean curvature hypersurfaces under suitable geometric assumptions in weighted warped products. Furthermore, we also prove very general Bernstein type results for the constant mean curvature equation for entire graphs in these ambient spaces.

scholarly journals Energy-minimizing maps from manifolds with nonnegative Ricci curvature

2019 ◽  
pp. 1950083 ◽  
Author(s):  
James Dibble
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Universal Covering ◽  
Complete Riemannian Manifold ◽  
Conjugate Points ◽  
Nonnegative Ricci Curvature ◽  
Universal Covering Space ◽  
Asymptotic Geometry ◽  
Bounded Below ◽  
Minimizing Maps

The energy of any [Formula: see text] representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant determined by the asymptotic geometry of the target, with equality if and only if the original map is totally geodesic. This conclusion also holds under the weaker assumption that the domain is finitely covered by a diffeomorphic product, and its universal covering space splits isometrically as a product with a flat factor, in a commutative diagram that follows from the Cheeger–Gromoll splitting theorem.

scholarly journals On Bernstein-Type Theorems in Semi-Riemannian Warped Products

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Wenjie Wang ◽  
Ximin Liu
Keyword(s):  
Mean Curvature ◽  
Warped Products ◽  
Bernstein Type ◽  
Complete Spacelike Hypersurfaces ◽  
Spacelike Hypersurfaces ◽  
The Mean

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).

scholarly journals The diameter of an immersed Riemannian manifold with bounded mean curvature

1981 ◽  
Vol 31 (2) ◽  
pp. 189-192 ◽  
Author(s):  
Th. Koufogiorgos ◽  
Ch. Baikoussis
Keyword(s):  
Riemannian Manifold ◽  
Isometric Immersion ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Complete Riemannian Manifold ◽  
Mean Curvature Vector ◽  
Nonpositive Sectional Curvature ◽  
The Mean ◽  
Simply Connected ◽  
Bounded Norm

AbstractLet M be an n-dimensional complete Riemannian manifold with Ricci curvature bounded from below. Let be an N-dimensional (N < n) complete, simply connected Riemannian manifold with nonpositive sectional curvature. We shall prove in this note that if there exists an isometric immersion φ of M into with the property that the immersed manifold is contained in a ball of radius R and that the mean curvature vector H of the immersion has bounded norm ∥H∥ > H0, (H0 > 0) then R > H−10.

scholarly journals EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN

2010 ◽  
Vol 53 (2) ◽  
pp. 321-332 ◽  
Author(s):  
SUN HEJUN ◽  
QI XUERONG
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Eigenvalue Problem ◽  
Unit Sphere ◽  
Mean Curvature ◽  
Complete Riemannian Manifold ◽  
Quadratic Polynomial ◽  
Polynomial Operator ◽  
Eigenvalue Estimates ◽  
The Mean

AbstractFor a bounded domain Ω in a complete Riemannian manifold M, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator Δ2 − aΔ + b of the Laplacian Δ, where a and b are the nonnegative constants. We obtain an inequality for eigenvalues which contains a constant that only depends on the mean curvature of M. It yields an upper bound of the (k + 1)th eigenvalue Λk + 1. As their applications, some inequalities and bounds of eigenvalues on a complete minimal submanifold in a Euclidean space and a unit sphere are obtained.

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