Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T15:03:51.375Z Has data issue: false hasContentIssue false
  • English
  • Français

Codimension two spacelike submanifolds of the Lorentz-Minkowski spacetime into the light cone

Part of: Local differential geometry Global differential geometry

Published online by Cambridge University Press:  17 January 2019

Luis J. Alías ,
Verónica L. Cánovas  and
Marco Rigoli
Luis J. Alías
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain ([email protected]; [email protected])
Verónica L. Cánovas
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain ([email protected]; [email protected])
Marco Rigoli
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, I-20133, Milano, Italy ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Following an original idea of Palmas, Palomo and Romero, recently developed in [12], we study codimension two spacelike submanifolds contained in the light cone of the Lorentz-Minkowski spacetime through an approach which allows us to compute their extrinsic and intrinsic geometries in terms of a single function u. As the first application of our approach, we classify the totally umbilical ones. For codimension two compact spacelike submanifolds into the light cone, we show that they are conformally diffeomorphic to the round sphere and that they are given by an explicit embedding written in terms of u. In the last part of the paper, we consider the case where the submanifold is (marginally, weakly) trapped. In particular, we derive some non-existence results for weakly trapped submanifolds into the light cone.

Keywords

spacelike submanifoldsLorentz-Minkowski spacetimelight conemean curvaturetrapped submanifolds

MSC classification

Primary: 53C40: Global submanifolds 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Secondary: 53B30: Lorentz metrics, indefinite metrics 53C50: Lorentz manifolds, manifolds with indefinite metrics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1523 - 1553
Copyright
Copyright © Royal Society of Edinburgh 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alías, L. J., Estudillo, F. J. M. and Romero, A.. Spacelike submanifolds with parallel mean curvature in pseudo-Riemannian space forms. Tsukuba J. Math. 21 (1997), 169–179.Google Scholar
2Alías, L. J., Mastrolia, P. and Rigoli, M.. Maximum principles and geometric applications. Springer Monographs in Mathematics (Cham: Springer, 2016).Google Scholar
3Asperti, A. C. and Dajczer, M.. Conformally flat Riemannian manifolds as hypersurfaces of the light cone. Canad. Math. Bull. 32 (1989), 281–285.Google Scholar
4Aviles, P. and McOwen, R. C.. Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds. J.Differ. Geom. 27 (1988), 225–239.Google Scholar
5Chruściel, P. T., Galloway, G. J. and Pollack, D.. Mathematical general relativity: a sampler. Bull. Amer. Math. Soc. (N.S.) 47 (2010), 567–638.Google Scholar
6Dajczer, M.. Submanifolds and isometric immersions. Mathematics Lecture Series,vol. 13 (Houston, TX: Publish or Perish, Inc., 1990).Google Scholar
7do Carmo, M. P.. Riemannian geometry. Mathematics: theory and applications (Boston, MA: Birkhauser Boston, Inc., 1992).10.1007/978-1-4757-2201-7Google Scholar
8Izumiya, S., Pei, D. and Romero Fuster, M. C.. Umbilicity of space-like submanifolds of Minkowski space. Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 375–387.Google Scholar
9Mars, M.. and Senovilla, J. M. M.. Trapped surfaces and symmetries. Class. Quantum Grav. 20 (2003), 129–1300.Google Scholar
10Mastrolia, P., Rigoli, M. and Setti, A. G.. Yamabe-type equations on complete, noncompact manifolds. Progress in Mathematics, vol. 302 (Basel AG, Basel: Birkhäuser/Springer, 2012).Google Scholar
11Navarro, M., Palmas, O. and Solis, D. A.. On the geometry of null hypersurfaces in Minkowski space. J. Geom. Phys. 75 (2014), 199–212.10.1016/j.geomphys.2013.10.005Google Scholar
12Palmas, O., Palomo, F. J. and Romero, A.. On the total mean curvature of a compact space-like submanifold in Lorentz-Minkowski spacetime. Proc. Roy. Soc. Edinburgh Sect. A 148 (2018), 199–210.Google Scholar
13Palomo, F. J. and Romero, A.. On spacelike surfaces in four-dimensional Lorentz-Minkowski spacetime through a light cone. Proc. Royal Soc. Edinburgh Sect. A 143 (2013), 881–892.Google Scholar
14Penrose, R.. Gravitational collapse and space-time singularities. Phys. Rev. Lett. 14 (1965), 57.Google Scholar
15Pigola, S., Rigoli, M. and Setti, A. G.. A remark on the maximum principle and stochastic completeness. Proc. Amer. Math. Soc. 131 (2003), 1283–1288.10.1090/S0002-9939-02-06672-8Google Scholar
16Pigola, S., Rigoli, M. and Setti, A. G.. Maximum principles on Riemannian manifolds and applications. Memoirs Amer. Math. Soc. 174 (2005), x+99 pp.Google Scholar