Compact graphs over a sphere of constant second order mean curvature

Barros, A.; Sousa, P.

doi:10.1090/s0002-9939-09-09862-1

Cited by 15 publications

(9 citation statements)

References 15 publications

(12 reference statements)

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“…In what follows we need a formula first derived in [2]. As stated below, it is the Lorentz version of the one stated and proved in [8].…”

Section: R−stability Of Spacelike Hypersurfacesmentioning

confidence: 98%

“…Here, motivated by the works [7] and [11], we consider spacelike hypersurfaces with constant r-th mean curvature in GRW spacetimes of constant sectional curvature in order to classify the strongly r-stable ones. For this, we will use a formula due to Barros and Sousa [8] for a operator L r , naturally attached to the operators P r that can be defined using the r-th mean curvatures, for a suitable support function. More precisely, we will prove the following result: be a closed strongly r-stable spacelike hypersurface.…”

Section: Introductionmentioning

confidence: 99%

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On the r-stability of spacelike hypersurfaces

Camargo

Caminha

Silva

et al. 2010

Journal of Geometry and Physics

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“…In what follows we need a formula first derived in [2]. As stated below, it is the Lorentz version of the one stated and proved in [8].…”

Section: R−stability Of Spacelike Hypersurfacesmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

On the r-stability of spacelike hypersurfaces

Camargo

Caminha

Silva

et al. 2010

Journal of Geometry and Physics

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show abstract

“…Although this operator appeared in geometry many years ago (see, e.g., [21,29]), there is a continues increase of applications of this operator in different areas of geometry in the last years (see, among others, [1,2,3,8,10,17,18,23,24,25,28]). …”

Section: Introductionmentioning

confidence: 99%

Generalized Newton transformation and its applications to extrinsic geometry

Andrzejewski

Kozłowski

Niedziałomski

2016

Asian Journal of Mathematics

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Abstract. In this article we introduce a generalization of the Newton transformation to the case of a system of endomorphisms. We show that it can be used in the context of extrinsic geometry of foliations and distributions yielding new integral formulas containing generalized extrinsic curvatures. IntroductionAnalyzing the study of Riemannian geometry we see that its basic concepts are related with some operators, such as shape, Ricci, Schouten operator, etc. and functions constructed of them, such as mean curvature, scalar curvature, Gauss-Kronecker curvature, etc. The most natural and useful functions are the ones derived from algebraic invariants of these operators e.g. by taking trace, determinant and in general the r-th symmetric functions σ r . However, the case r > 1 is strongly nonlinear and therefore more complicated. The powerful tool to deal with this problem is the Newton transformation T r of an endomorphism A (strictly related with the Newton's identities) which, in a sense, enables a linearization of σ r , (r + 1)σ r+1 = tr (AT r ).Although this operator appeared in geometry many years ago (see, e.g., [21,29]), there is a continues increase of applications of this operator in different areas of geometry in the last years (see, among others, [1,2,3,8,10,17,18,23,24,25,28]).All these results cause a natural question, what happen if we have a family of operators i.e. how to define the Newton transformation for a family of endomorphisms. A partial answer to this question can be found in the literature (operator T r and the scalar S r for even r [5,15]), nevertheless, we expect that this case is much more subtle. This is because in the case of family of operators we should obtain more natural functions as in the case of one and consequently more information about geometry. In order to do this, for any multi-index u 2000 Mathematics Subject Classification. 53C12; 53C65.

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“…On the other hand, many authors (see, among the others, [1,3,6,8,12,16]) investigated recently higher-order mean curvatures of hypersurfaces using the Newton transformations T r of the shape operator. In this article, we show that these transformations can also be applied successfully for foliations.…”

Section: Introductionmentioning

confidence: 99%

The Newton transformation and new integral formulae for foliated manifolds

Andrzejewski

Walczak

2009

Ann Glob Anal Geom

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In this article, we show that the Newton transformations of the shape operator can be applied successfully to foliated manifolds. Using these transformations, we generalize known integral formulae (due to Brito-Langevin-Rosenberg, Ranjan, Walczak, etc.) for foliations of codimension one. We obtain integral formulae involving r th mean curvature of the second fundamental form of a foliation, the Jacobi operator in the direction orthogonal to the foliation, and their products. We apply our formulae to totally umbilical foliations and foliations whose leaves have constant second order mean curvature.

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Compact graphs over a sphere of constant second order mean curvature

Cited by 15 publications

References 15 publications

On the r-stability of spacelike hypersurfaces

On the r-stability of spacelike hypersurfaces

Generalized Newton transformation and its applications to extrinsic geometry

The Newton transformation and new integral formulae for foliated manifolds

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