Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A NOTE ON MAXIMAL HYPERSURFACES IN A GENERALIZED ROBERTSON-WALKER SPACETIME

  • Received : 2021.07.01
  • Accepted : 2021.10.14
  • Published : 2022.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this note, we apply a maximum principle related to volume growth of a complete noncompact Riemannian manifold, which was recently obtained by Alías, Caminha and do Nascimento in [4], to establish new uniqueness and nonexistence results concerning maximal spacelike hypersurfaces immersed in a generalized Robertson-Walker (GRW) spacetime obeying the timelike convergence condition. A study of entire solutions for the maximal hypersurface equation in GRW spacetimes is also made and, in particular, a new Calabi-Bernstein type result is presented.

Keywords

References

  1. A. L. Albujer, New examples of entire maximal graphs in ℍ2 × ℝ1, Differential Geom. Appl. 26 (2008), no. 4, 456-462. https://doi.org/10.1016/j.difgeo.2007.11.035
  2. A. L. Albujer and L. J. Alias, Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces, J. Geom. Phys. 59 (2009), no. 5, 620-631. https://doi.org/10.1016/j.geomphys.2009.01.008
  3. J. A. Aledo, L. J. Alias, and A. Romero, Integral formulas for compact space-like hypersurfaces in de Sitter space: applications to the case of constant higher order mean curvature, J. Geom. Phys. 31 (1999), no. 2-3, 195-208. https://doi.org/10.1016/S0393-0440(99)00008-X
  4. L. J. Alias, A. Caminha, and F. Y. do Nascimento, A maximum principle related to volume growth and applications, Ann. Mat. Pura Appl. (4) 200 (2021), no. 4, 1637-1650. https://doi.org/10.1007/s10231-020-01051-9
  5. L. J. Alias, A. G. Colares, and H. F. de Lima, On the rigidity of complete spacelike hypersurfaces immersed in a generalized Robertson-Walker spacetime, Bull. Braz. Math. Soc. (N.S.) 44 (2013), no. 2, 195-217. https://doi.org/10.1007/s00574-013-0009-7
  6. L. J. Alias and S. Montiel, Uniqueness of spacelike hypersurfaces with constant mean curvature in generalized Robertson-Walker spacetimes, in Differential geometry, Valencia, 2001, 59-69, World Sci. Publ., River Edge, NJ, 2002.
  7. L. J. Alias, A. Romero, and M. Sanchez, Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes, Gen. Relativity Gravitation 27 (1995), no. 1, 71-84. https://doi.org/10.1007/BF02105675
  8. J. K. Beem, P. E. Ehrlich, and K. L. Easley, Global Lorentzian geometry, second edition, Monographs and Textbooks in Pure and Applied Mathematics, 202, Marcel Dekker, Inc., New York, 1996.
  9. M. Caballero, A. Romero, and R. M. Rubio, Uniqueness of maximal surfaces in generalized Robertson-Walker spacetimes and Calabi-Bernstein type problems, J. Geom. Phys. 60 (2010), no. 3, 394-402. https://doi.org/10.1016/j.geomphys.2009.11.008
  10. H. F. de Lima, F. R. dos Santos, and J. G. Araujo, Solutions to the maximal spacelike hypersurface equation in generalized Robertson-Walker spacetimes, Electron. J. Differential Equations 2018, Paper No. 80, 14 pp.
  11. H. F. de Lima and U. L. Parente, On the geometry of maximal spacelike hypersurfaces immersed in a generalized Robertson-Walker spacetime, Ann. Mat. Pura Appl. (4) 192 (2013), no. 4, 649-663. https://doi.org/10.1007/s10231-011-0241-y
  12. S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London, 1973.
  13. T. Ishihara, Maximal spacelike submanifolds of a pseudo-Riemannian space of constant curvature, Michigan Math. J. 35 (1988), no. 3, 345-352. https://doi.org/10.1307/mmj/1029003815
  14. J. M. Latorre and A. Romero, Uniqueness of noncompact spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes, Geom. Dedicata 93 (2002), 1-10. https://doi.org/10.1023/A:1020341512060
  15. S. Montiel, Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes, Math. Ann. 314 (1999), no. 3, 529-553. https://doi.org/10.1007/s002080050306
  16. B. O'Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983.
  17. J. A. S. Pelegrin, A. Romero, and R. M. Rubio, Uniqueness of complete maximal hypersurfaces in spatially open (n + 1)-dimensional Robertson-Walker spacetimes with flat fiber, Gen. Relativity Gravitation 48 (2016), no. 6, Art. 70, 14 pp. https://doi.org/10.1007/s10714-016-2069-7
  18. M. Rainer and H.-J. Schmidt, Inhomogeneous cosmological models with homogeneous inner hypersurface geometry, Gen. Relativity Gravitation 27 (1995), no. 12, 1265-1293. https://doi.org/10.1007/BF02153317
  19. A. Romero and R. M. Rubio, On maximal surfaces in certain non-flat 3-dimensional Robertson-Walker spacetimes, Math. Phys. Anal. Geom. 15 (2012), no. 3, 193-202. https://doi.org/10.1007/s11040-012-9108-8
  20. A. Romero, R. M. Rubio, and J. J. Salamanca, Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson-Walker spacetimes, Classical Quantum Gravity 30 (2013), no. 11, 115007, 13 pp. https://doi.org/10.1088/0264-9381/30/11/115007
  21. A. Romero, R. M. Rubio, and J. J. Salamanca, A new approach for uniqueness of complete maximal hypersurfaces in spatially parabolic GRW spacetimes, J. Math. Anal. Appl. 419 (2014), no. 1, 355-372. https://doi.org/10.1016/j.jmaa.2014.04.063