On the first eigenvalue of spacelike hypersurfaces in Lorentzian space

Bing Ye Wu

Archivum Mathematicum (2006)

  • Volume: 042, Issue: 3, page 233-238
  • ISSN: 0044-8753

Abstract

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In this paper we obtain a lower bound for the first Dirichlet eigenvalue of complete spacelike hypersurfaces in Lorentzian space in terms of mean curvature and the square length of the second fundamental form. This estimate is sharp for totally umbilical hyperbolic spaces in Lorentzian space. We also get a sufficient condition for spacelike hypersurface to have zero first eigenvalue.

How to cite

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Wu, Bing Ye. "On the first eigenvalue of spacelike hypersurfaces in Lorentzian space." Archivum Mathematicum 042.3 (2006): 233-238. <http://eudml.org/doc/249804>.

@article{Wu2006,
abstract = {In this paper we obtain a lower bound for the first Dirichlet eigenvalue of complete spacelike hypersurfaces in Lorentzian space in terms of mean curvature and the square length of the second fundamental form. This estimate is sharp for totally umbilical hyperbolic spaces in Lorentzian space. We also get a sufficient condition for spacelike hypersurface to have zero first eigenvalue.},
author = {Wu, Bing Ye},
journal = {Archivum Mathematicum},
keywords = {Lorentzian space; spacelike hypersurface; the first eigenvalue; Gauss map; Lorentzian space; spacelike hypersurface; the first eigenvalue; Gauss map},
language = {eng},
number = {3},
pages = {233-238},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the first eigenvalue of spacelike hypersurfaces in Lorentzian space},
url = {http://eudml.org/doc/249804},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Wu, Bing Ye
TI - On the first eigenvalue of spacelike hypersurfaces in Lorentzian space
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 3
SP - 233
EP - 238
AB - In this paper we obtain a lower bound for the first Dirichlet eigenvalue of complete spacelike hypersurfaces in Lorentzian space in terms of mean curvature and the square length of the second fundamental form. This estimate is sharp for totally umbilical hyperbolic spaces in Lorentzian space. We also get a sufficient condition for spacelike hypersurface to have zero first eigenvalue.
LA - eng
KW - Lorentzian space; spacelike hypersurface; the first eigenvalue; Gauss map; Lorentzian space; spacelike hypersurface; the first eigenvalue; Gauss map
UR - http://eudml.org/doc/249804
ER -

References

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  1. Cheung L. F., Leung P. F., Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space, Math. Z. 236 (2001), 525–530. Zbl0990.53029MR1821303
  2. Cheng S. Y., Yau S. T., Differential equations on Riemannian manifolds and geometric applications, Comm. Pure Appl. Math. 28 (1975), 333–354. (1975) MR0385749
  3. Kobayashi S., Nomizu K., Foundations of Differential Geometry, vol II, Interscience, New York, 1969. (1969) Zbl0175.48504MR0238225
  4. Mckean H. P., An upper bound for the spectrum of Δ on a manifold of negative curvature, J. Differential Geometry 4 (1970), 359–366. (1970) MR0266100
  5. Pacellibessa G., Montenegro J. F., Eigenvalue estimates for submanifolds with locally bounded mean curvature, Ann. Glob. Anal. Geom. 24 (2003), 279–290. MR1996771
  6. Schoen R., Yau S. T., Lectures on differential geometry, Lecture Notes in Geom. Topo. 1 (1994). (1994) Zbl0830.53001MR1333601

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