Embedded positive constant r-mean curvature hypersurfaces in Mm × R

Cheng, Xu; Rosenberg, Harold

doi:10.1590/s0001-37652005000200001

Cited by 39 publications

(27 citation statements)

References 7 publications

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“…The study of hypersurfaces with constant higher-order mean curvatures has been of increasing interest in the last years (see, among the others, [2,12,15,16]). Now, we provide some results for foliations whose leaves have constant S 2 .…”

Section: Foliations Whose Leaves Have Constant Smentioning

confidence: 99%

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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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Section: Foliations Whose Leaves Have Constant Smentioning

confidence: 99%

“…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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“…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%

“…Using integration on these bundles we define generalized mean curvatures σ u (see (16)) for distributions and total extrinsic curvatures σ M u (see (17)). Moreover, we define a new set of global vector fields Y u generalizing (2), obtained from sections Y u (see (18)), by integrating over the fibers of P .…”

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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“…To establish our estimate, we apply the technique used by Cheng and Rosenberg to prove Theorem 4.1 in [10]. There, they obtain such an estimate concerning a compact vertical graph n immersed with positive constant r -mean curvature into a Riemannian product R × M n and whose boundary ∂ is contained into the slice {0} × M n (we note that Rosenberg [20] already obtained this type of estimate concerning such graphs in the Riemannian space forms; see also Hoffman et al [14] for the case when the ambient is a product R × M 2 , where M 2 is a Riemannian surface).…”

Section: Introductionmentioning

confidence: 99%

Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces

Colares

Lima

2008

Gen Relativ Gravit

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In this paper we obtain a height estimate concerning compact space-like hypersurfaces n immersed with some positive constant r -mean curvature into an (n + 1)-dimensional Lorentzian product space −R × M n , and whose boundary is contained into a slice {t} × M n . By considering the hyperbolic caps of the LorentzMinkowski space L n+1 , we show that our estimate is sharp. Furthermore, we apply this estimate to study the complete space-like hypersurfaces immersed with some positive constant r -mean curvature into a Lorentzian product space. For instance, when the ambient space-time is spatially closed, we show that such hypersurfaces must satisfy the topological property of having more than one end which constitutes a necessary condition for their existence.

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Embedded positive constant r-mean curvature hypersurfaces in Mm × R

Cited by 39 publications

References 7 publications

The Newton transformation and new integral formulae for foliated manifolds

The Newton transformation and new integral formulae for foliated manifolds

Generalized Newton transformation and its applications to extrinsic geometry

Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces

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