Hypersurfaces in pseudo-Euclidean spaces satisfying a linear condition on the linearized operator of a higher order mean curvature

We study the problem of lightlike hypersurface immersed into Robertson-Walker (RW) spacetimes in this paper, where the screen bundle of the hypersurface has constant higher order mean curvature. We consider the following question: under what conditions is the compact lightlike hypersurface totally umbilical? Our approach is based on the relationship between the lightlike hypersurface with its screen bundle and the Minkowski formulae for the screen bundle.

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Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004107000576 ◽

2007 ◽

Vol 143 (3) ◽

pp. 703-729 ◽

Cited By ~ 56

Author(s):

LUIS J. ALÍAS ◽

A. GERVASIO COLARES

Keyword(s):

Mean Curvature ◽

Differential Operators ◽

Spacelike Hypersurface ◽

Convergence Condition ◽

Higher Order ◽

Spacelike Hypersurfaces ◽

Newton Transformations ◽

Higher Order Mean Curvature ◽

Associated Differential Operators

AbstractIn this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so callednull convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.

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A note on Weingarten hypersurfaces in the warped product manifold

International Journal of Mathematics ◽

10.1142/s0129167x14501213 ◽

2014 ◽

Vol 25 (14) ◽

pp. 1450121 ◽

Cited By ~ 3

Author(s):

Haizhong Li ◽

Yong Wei ◽

Changwei Xiong

Keyword(s):

Mean Curvature ◽

Warped Product ◽

Constant Mean Curvature ◽

Higher Order ◽

Product Manifold ◽

Weingarten Surfaces ◽

Warped Product Manifold ◽

Higher Order Mean Curvature ◽

Special Case ◽

Differential Geom

In this paper, we consider the closed embedded hypersurface Σ in the warped product manifold [Formula: see text] equipped with the metric g = dr2 + λ(r)2 gN. We give some characterizations of slice {r} × N by the condition that Σ has constant weighted higher-order mean curvatures (λ′)αpk, or constant weighted higher-order mean curvature ratio (λ′)αpk/p1, which generalize Brendle's [Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 247–269] and Brendle–Eichmair's [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] results. In particular, we show that the assumption convex of Brendle–Eichmair's result [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] is unnecessary. Here pk is the kth normalized mean curvature of the hypersurface Σ. As a special case, we also give some characterizations of geodesic spheres in ℝn, ℍn and [Formula: see text], which generalize the classical Alexandrov-type results.

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Erratum to “Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes”. Math. Proc. Camb. Phil. Soc. (2012), 152, 365–383.

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004113000376 ◽

2013 ◽

Vol 155 (2) ◽

pp. 375-377

Author(s):

LUIS J. ALÍAS ◽

DEBORA IMPERA ◽

MARCO RIGOLI

Keyword(s):

Mean Curvature ◽

Higher Order ◽

Spacelike Hypersurfaces ◽

Higher Order Mean Curvature ◽

Formula Group ◽

Image Position ◽

Correct Expression

The proof of Corollary 4⋅3 in our paper [1] is not correct because there is a mistake in the expression given for ∥X* ∧ Y*∥2 on page 374. In fact, the correct expression for this term is \begin{eqnarray*} \norm{X^*\wedge Y^*}^2 & = & \norm{X^*}^2\norm{Y^*}^2-\pair{X^*,Y^*}^2\\ {} & = & 1+\pair{X,T}^2+\pair{Y,T}^2\geq 1, \end{eqnarray*} and then the inequality (4⋅9) is no longer true. Observe that all the previous reasoning before the wrong expression for ∥X* ∧ Y*∥2 is correct.

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