A Priori Estimates of the Principal Curvatures of Spacelike Hypersurfaces in de Sitter Space with Applications to Hypersurfaces in Hyperbolic Space

1992 ◽  
Vol 114 (3) ◽  
pp. 605 ◽  
Author(s):  
Vladimir Oliker
Keyword(s):  
Hyperbolic Space ◽  
A Priori Estimates ◽  
A Priori ◽  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Priori Estimates ◽  
Spacelike Hypersurfaces

scholarly journals Weingarten spacelike hypersurfaces in a de Sitter space

2012 ◽  
Vol 20 (1) ◽  
pp. 387-406
Author(s):  
Junfeng Chen ◽  
Shichang Shu
Keyword(s):  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Riemannian Product ◽  
Spacelike Hypersurfaces ◽  
Hyperbolic Cylinder

Abstract We study some Weingarten spacelike hypersurfaces in a de Sitter space S1n+1 (1). If the Weingarten spacelike hypersurfaces have two distinct principal curvatures, we obtain two classification theorems which give some characterization of the Riemannian product Hk(1−coth2 ϱ)× Sn−k(1 − tanh2 ϱ), 1 < k < n − 1 in S1n+1(1), the hyperbolic cylinder H1(1 − coth2 ϱ) × Sn-1(1 − tanh2 ϱ) or spherical cylinder Hn−1(1 − coth2 ϱ) × S1(1 − tanh2 ϱ) in S1n+1 (1)

Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space

2019 ◽  
Vol 69 (6) ◽  
pp. 1437-1446
Author(s):  
Rong Mi
Keyword(s):  
Lorentz Space ◽  
De Sitter Space ◽  
Sao Paulo ◽  
Maximum Principles ◽  
Isoparametric Hypersurface ◽  
De Sitter ◽  
Principal Curvatures ◽  
Totally Umbilical ◽  
Type Formula ◽  
Spacelike Hypersurfaces

Abstract Our purpose in this paper is to study complete linear Weingarten space-like hypersurface immersed in locally symmetric Lorentz space obeying some curvature conditions. Our approach is based on the use of a Simons type formula related to an appropriated Cheng-Yau modified operator jointly with some generalized maximum principles, we obtain that such a space-like hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. This result corresponds to a natural improvement of previous ones due to de Lima, dos Santos, Velásquez [On the umbilicity of complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space, São Paulo J. Math. Sci. 11 (2017), 456–470] and Alías, de Lima, dos Santos [New characterizations of linear Weingarten spacelike hypersurfaces in de Sitter space, Pacific J. Math. 292 (2018), 1–19].

Complete spacelike hypersurfaces in a de Sitter space

2006 ◽  
Vol 73 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Shu Shichang
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Scalar Curvature ◽  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Complete Spacelike Hypersurfaces ◽  
Spacelike Hypersurfaces ◽  
The Mean

In this paper, we characterise the n-dimensional (n ≥ 3) complete spacelike hypersurfaces Mn in a de Sitter space with constant scalar curvature and with two distinct principal curvatures. We show that if the multiplicities of such principal curvatures are greater than 1, then Mn is isometric to Hk (sinh r) × Sn−k (cosh r), 1 < k < n − 1. In particular, when Mn is the complete spacelike hypersurfaces in with the scalar curvature and the mean curvature being linearly related, we also obtain a characteristic Theorem of such hypersurfaces.

scholarly journals Compact Spacelike Hypersurfaces with Constant Mean Curvature in the Antide Sitter Space

2009 ◽  
Vol 2009 ◽  
pp. 1-12
Author(s):  
Henrique F. de Lima ◽  
Joseilson R. de Lima
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Spacelike Hypersurface ◽  
De Sitter Space ◽  
De Sitter ◽  
Height Estimate ◽  
Spacelike Hypersurfaces ◽  
Hyperbolic Domains

We obtain a height estimate concerning to a compact spacelike hypersurfaceΣnimmersed with constant mean curvatureHin the anti-de Sitter spaceℍ1n+1, when its boundary∂Σis contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic spaceℍn. Our estimate depends only on the value ofHand on the geometry of∂Σ.As applications of our estimate, we obtain a characterization of hyperbolic domains ofℍ1n+1and nonexistence results in connection with such types of hypersurfaces.

ON COMPLETE SPACELIKE HYPERSURFACES WITH CONSTANT m-TH MEAN CURVATURE IN AN ANTI-DE SITTER SPACE

2010 ◽  
Vol 21 (05) ◽  
pp. 551-569 ◽  
Author(s):  
B. Y. WU
Keyword(s):  
Mean Curvature ◽  
Constant Scalar Curvature ◽  
De Sitter Space ◽  
Second Fundamental Form ◽  
De Sitter ◽  
Principal Curvatures ◽  
Global Rigidity ◽  
Complete Spacelike Hypersurfaces ◽  
Spacelike Hypersurfaces ◽  
Hyperbolic Cylinders

We investigate complete spacelike hypersurfaces in an Anti-de Sitter space with constant m-th mean curvature and two distinct principal curvatures. By using Otsuki's idea, we obtain some global classification results. For their application, we obtain some characterizations for hyperbolic cylinders. We prove that the only complete spacelike hypersurfaces in Anti-de Sitter (n + 1)-spaces (n ≥ 3) of constant mean curvature or constant scalar curvature with two distinct principal curvatures λ and μ satisfying inf (λ - μ)2 > 0 are the hyperbolic cylinders. It is a little surprising that the corresponding result does not hold for m-th mean curvature when m > 2. We also obtain some global rigidity results for hyperbolic cylinders in terms of square length of the second fundamental form.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

2020 ◽  
Vol 57 (1) ◽  
pp. 68-90 ◽  
Author(s):  
Tahir S. Gadjiev ◽  
Vagif S. Guliyev ◽  
Konul G. Suleymanova
Keyword(s):  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Morrey Spaces ◽  
Smooth Bounded Domain ◽  
Weighted Morrey Spaces ◽  
Priori Estimates ◽  
Muckenhoupt Class ◽  
Morrey Estimates ◽  
Dirichlet Boundary Problem

Abstract In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Convergence of the Rosenau-Korteweg-de Vries Equation to the Korteweg-de Vries One

2020 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
A Priori Estimates ◽  
Vries Equation ◽  
A Priori ◽  
Diffusion Parameter ◽  
Priori Estimates ◽  
De Vries ◽  
Wall Interactions

The Rosenau-Korteweg-de Vries equation describes the wave-wave and wave-wall interactions. In this paper, we prove that, as the diffusion parameter is near zero, it coincides with the Korteweg-de Vries equation. The proof relies on deriving suitable a priori estimates together with an application of the Aubin-Lions Lemma.

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