New characterizations of spacelike hyperplanes in the steady state space

  • Cícero P. Aquino
  • Halyson I. Baltazar
  • Henrique F. de Lima

Abstract

In this article, we deal with complete spacelike hypersurfaces immersed in an open region of the de Sitter space $\mathbb {S}^{n+1}_{1}$ which is known as the steady state space $\mathcal {H}^{n+1}$. Under suitable constraints on the behavior of the higher order mean curvatures of these hypersurfaces, we are able to prove that they must be spacelike hyperplanes of $\mathcal {H}^{n+1}$. Furthermore, through the analysis of the hyperbolic cylinders of $\mathcal {H}^{n+1}$, we discuss the importance of the main hypothesis in our results. Our approach is based on a generalized maximum principle at infinity for complete Riemannian manifolds.

References

Albujer, A. L. and Al\'ıas, L. J., Spacelike hypersurfaces with constant mean curvature in the steady state space, Proc. Amer. Math. Soc. 137 (2009), no. 2, 711–721. https://doi.org/10.1090/S0002-9939-08-09546-4

Al\'ıas, L. J., Brasil, Jr., A., and Gervasio Colares, A., Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications, Proc. Edinb. Math. Soc. (2) 46 (2003), no. 2, 465–488. https://doi.org/10.1017/S0013091502000500

Al\'ıas, L. J. and Colares, A. G., Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 3, 703–729. https://doi.org/10.1017/S0305004107000576

Al\'ıas, L. J., de Lira, J. H. S., and Malacarne, J. M., Constant higher-order mean curvature hypersurfaces in Riemannian spaces, J. Inst. Math. Jussieu 5 (2006), no. 4, 527–562. https://doi.org/10.1017/S1474748006000077

Andersson, L., Galloway, G. J., and Howard, R., A strong maximum principle for weak solutions of quasi-linear elliptic equations with applications to Lorentzian and Riemannian geometry, Comm. Pure Appl. Math. 51 (1998), no. 6, 581–624. https://doi.org/10.1002/(SICI)1097-0312(199806)51:6<581::AID-CPA2>3.3.CO;2-E

Aquino, C. P., de Lima, H. F., dos Santos, F. R., and Velásquez, M. A. L., Characterizations of spacelike hyperplanes in the steady state space via generalized maximum principles, Milan J. Math. 83 (2015), no. 2, 199–209. https://doi.org/10.1007/s00032-015-0238-x

Barbosa, J. L. M. and Colares, A. G., Stability of hypersurfaces with constant $r$-mean curvature, Ann. Global Anal. Geom. 15 (1997), no. 3, 277–297. https://doi.org/10.1023/A:1006514303828

Bondi, H. and Gold, T., On the generation of magnetism by fluid motion, Monthly Not. Roy. Astr. Soc. 110 (1950), 607–611. https://doi.org/10.1093/mnras/110.6.607

Caminha, A., The geometry of closed conformal vector fields on Riemannian spaces, Bull. Braz. Math. Soc. (N.S.) 42 (2011), no. 2, 277–300. https://doi.org/10.1007/s00574-011-0015-6

Caminha, A. and de Lima, H. F., Complete vertical graphs with constant mean curvature in semi-Riemannian warped products, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 1, 91–105.

Colares, A. G. and de Lima, H. F., Spacelike hypersurfaces with constant mean curvature in the steady state space, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 2, 287–302.

Colares, A. G. and de Lima, H. F., On the rigidity of spacelike hypersurfaces immersed in the steady state space $\mathcal H^n+1$, Publ. Math. Debrecen 81 (2012), no. 1-2, 103–119. https://doi.org/10.5486/PMD.2012.5083

Eschenburg, J.-H., Maximum principle for hypersurfaces, Manuscripta Math. 64 (1989), no. 1, 55–75. https://doi.org/10.1007/BF01182085

Gaffney, M. P., A special Stokes's theorem for complete Riemannian manifolds, Ann. of Math. (2) 60 (1954), 140–145. https://doi.org/10.2307/1969703

Hawking, S. W. and Ellis, G. F. R., The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, no. 1, Cambridge University Press, Cambridge, 1973.

Hoyle, F., A new model for the expanding universe, Monthly Not. Roy. Astr. Soc. 108 (1948), no. 5, 372–382. https://doi.org/10.1093/mnras/108.5.372

de Lima, H. F., Spacelike hypersurfaces with constant higher order mean curvature in de Sitter space, J. Geom. Phys. 57 (2007), no. 3, 967–975. https://doi.org/10.1016/j.geomphys.2006.07.005

Montiel, S., An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J. 37 (1988), no. 4, 909–917. https://doi.org/10.1512/iumj.1988.37.37045

Montiel, S., A characterization of hyperbolic cylinders in the de Sitter space, Tôhoku Math. J. (2) 48 (1996), no. 1, 23–31. https://doi.org/10.2748/tmj/1178225410

Montiel, S., Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, Indiana Univ. Math. J. 48 (1999), no. 2, 711–748. https://doi.org/10.1512/iumj.1999.48.1562

Montiel, S., Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces, J. Math. Soc. Japan 55 (2003), no. 4, 915–938. https://doi.org/10.2969/jmsj/1191418756

Rosenberg, H., Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), no. 2, 211–239.

Sachs, R. K. and Wu, H. H., General relativity for mathematicians, Graduate Texts in Mathematics, vol. 48, Springer-Verlag, New York-Heidelberg, 1977.

Weinberg, S., Gravitation and cosmology: principles and applications of the general theory of relativity, John Wiley & Sons, New York, 1972.

Yau, S. T., Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. https://doi.org/10.1512/iumj.1976.25.25051
Published
2020-03-29
How to Cite
Aquino, C., Baltazar, H., & de Lima, H. (2020). New characterizations of spacelike hyperplanes in the steady state space. MATHEMATICA SCANDINAVICA, 126(1), 61-72. https://doi.org/10.7146/math.scand.a-117703
Section
Articles