Marginally trapped submanifolds in space forms with arbitrary signature

Anciaux, Henri

doi:10.2140/pjm.2014.272.257

Cited by 8 publications

(4 citation statements)

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“…It is easily seen that V is in fact independent on the election of a local coordinate z and any particular conformal lift of ψ. So V is can be viewed as a vector sub-bundle V ⊂ R 5 1 × Σ on which the ambient metric of R 5 1 induces a vector bundle metric of signature (3,1). Each fiber V x determines a Moebius invariant 2-sphere…”

Section: Preliminariesmentioning

confidence: 99%

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Conformal geometry of marginally trapped surfaces in $$\mathbb {S}^4_1$$ S 1 4

Hulett

2016

Beitr Algebra Geom

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A spacelike surface S ⊂ S 4 1 is marginally trapped if its mean curvature vector is lightlike. On any oriented spacelike surface S ⊂ S 4 1 we show that a choice of orientation of the normal bundle ν(S) determines a smooth map G : S → S 3 which we call the null Gauss map of S. We show that if S is marginally trapped then G is a conformal immersion away the zeros of certain quadratic Hopf-differential of S and so the surface G(S) is uniquely determined up to conformal transformations of S 3 by two invariants: the normal Hopf differential κ and the Schwartzian derivative s. We show that these invariants plus an additional quadratic differential δ are related by a differential equation and determine the geometry of S up to ambient isometries of S 4 1 . This allows us to obtain a characterization of marginally trapped surfaces S whose null Gauss image is a constrained Willmore surface in S 3 [6]. As an application of these results we construct and study integrable non-trivial one-parameter deformations of marginally trapped surfaces with non-zero parallel mean curvature vector and those with flat normal bundle.2010 Mathematics Subject Classification. 53C42, 53C50, 53C43.

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Section: Preliminariesmentioning

confidence: 99%

“…Also [13] and [14] deal with a classifications of marginally trapped surfaces with parallel mean curvature. The notion of marginally trappedness has also been considered recently in higher dimensions and co-dimensions with very interesting results, see [3], [4].…”

Section: Introductionmentioning

confidence: 99%

Conformal geometry of marginally trapped surfaces in $$\mathbb {S}^4_1$$ S 1 4

Hulett

2016

Beitr Algebra Geom

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“…We note that marginally trapped submanifolds can also be regarded as the Lorentzian dual of minimal submanifolds, when compared with the Riemannian context. For a thorough discussion concerning marginally trapped submanifolds, we refer the readers to the works [4]- [6], [10], [11].…”

Section: Introductionmentioning

confidence: 99%

Weakly trapped submanifolds in standard static spacetimes

Freitas

Lima

et al. 2019

Arkiv För Matematik

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We study weakly trapped submanifolds of codimension two in a standard static spacetime. In this setting, we apply some generalized maximum principles in order to investigate the geometry of these trapped submanifolds. For instance, we establish sufficient conditions to guarantee that such a spacelike submanifold must be a hypersurface of the Riemannian base of the ambient spacetime. As a consequence, we prove that there do not exist n-dimensional compact (without boundary) trapped submanifolds immersed in an (n+2)-dimensional standard static spacetime. Such a nonexistence result was originally obtained for stationary spacetimes by Mars and Senovilla [20]. Furthermore, we also investigate parabolic weakly trapped submanifolds immersed in a standard static spacetime.

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“…Já na Seção 2.2 introduzimos os chamados Referenciais de Penrose (null frames) para subvariedades de codimensão 2, que são uma ferramenta poderosa para obter diversos resultados. Como exemplo de aplicação, seguimos Anciaux em [49] para mostrar que toda subvariedade de codimensão 2 com Segunda Forma Fundamental de tipo luz em M n+2 ν (c) está contida em um hiperplano de tipo luz. Na Seção 2.3, seguindo [46], introduzimos a noção de λ-isotropia, e estudamos as relações disto com umbilidade e pseudo-umbilicidade de superfícies em vários ambientes.…”

Section: Aprisionamento Versus Isotropiaunclassified

Caracterizações de subvariedades marginalmente aprisionadas em formas espaciais

Couto¹

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Marginally trapped submanifolds in space forms with arbitrary signature

Abstract: We give explicit representation formulas for marginally trapped submanifolds of codimension two in pseudo-Riemannian spaces with arbitrary signature and constant sectional curvature.

Cited by 8 publications

References 10 publications

Conformal geometry of marginally trapped surfaces in $$\mathbb {S}^4_1$$ S 1 4

Conformal geometry of marginally trapped surfaces in $$\mathbb {S}^4_1$$ S 1 4

Weakly trapped submanifolds in standard static spacetimes

Caracterizações de subvariedades marginalmente aprisionadas em formas espaciais

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