, Volume 63, Issue 2, pp 323–332 | Cite as

Height estimates and half-space type theorems in weighted product spaces with nonnegative Bakry–Émery–Ricci curvature

  • Henrique F. de Lima
  • Márcio S. Santos


We prove height estimates concerning compact hypersurfaces with nonzero constant weighted mean curvature and whose boundary is contained into a slice of a weighted product space of nonnegative Bakry–Émery–Ricci curvature. As applications of our estimates, we obtain half-space type results related to complete noncompact hypersurfaces properly immersed in such an ambient space.


Weighted product spaces Bakry–Émery–Ricci tensor Compact hypersurfaces Height estimates Half-space type theorems 

Mathematics Subject Classification

Primary 53C42 Secondary 53B30 53C50 



The first author is partially supported by CNPq, Brazil, Grant 303977/2015-9. The authors would like to thank the referee for giving valuable suggestions and comments which improved the paper.


  1. 1.
    Albujer, A.L., Aledo, J.A., Alías, L.J.: On the scalar curvature of hypersurfaces in spaces with a Killing field. Adv. Geom. 10, 487–503 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aledo, J.A., Espinar, J.M., Gálvez, J.A.: Surfaces with constant curvature in \({\mathbb{S}}^2\times {\mathbb{R}}\) and \({\mathbb{H}}^2\times {\mathbb{R}}\). Height estimates and representation. Bull. Braz. Math. Soc. 38, 533–554 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aledo, J.A., Espinar, J.M., Gálvez, J.A.: Height estimates for surfaces with positive constant mean curvature in \(M^2\times {\mathbb{R}}\). Ill. J. Math. 52, 203–211 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Alías, L.J., Dajczer, M.: Constant mean curvature hypersurfaces in warped product spaces. Proc. Edinb. Math. Soc. 50, 511–526 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Alías, L.J., Dajczer, M., Ripoll, J.: A Bernstein-type theorem for Riemannian manifolds with a Killing field. Ann. Glob. Anal. Geom 31, 363–373 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bakry, D., Émery, M.: Diffusions hypercontractives. In: Azéma, J., Yor, M. (eds.) Séminaire de Probabilités XIX, 1983/84. Lecture Notes in Mathematics, vol. 1123, pp. 177–206. Springer, Berlin (1985)Google Scholar
  7. 7.
    Case, J.S.: Singularity theorems and the Lorentzian splitting theorem for the Bakry–Émery–Ricci tensor. J. Geom. Phys. 60, 477–490 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cheng, X., Rosenberg, H.: Embedded positive constant \(r\)-mean curvature hypersurfaces in \(M^m\times {\mathbb{R}} \). Ann. Braz. Acad. Sci 77, 183–199 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Colares, A.G., de Lima, H.F.: Space-like hypersurfaces with positive constant \(r\)-mean curvature in Lorentzian product spaces. Gen. Relativ. Gravity 40, 2131–2147 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    de Lima, H.F.: A sharp height estimate for compact spacelike hypersurfaces with constant \(r\)-mean curvature in the Lorentz–Minkowski space and application. Differ. Geom. Appl. 26, 445–455 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Espinar, J.M., Rosenberg, H.: Complete constant mean curvaturesurfaces and Bernstein type theorems in \(M^2\times {\mathbb{R}}\). J. Differ. Geom 82, 611–628 (2009)CrossRefzbMATHGoogle Scholar
  12. 12.
    Fang, F., Li, X.-D., Zhang, Z.: Two generalizations of Cheeger–Gromoll splitting theorem via Bakry–Émery–Ricci curvature. Ann. Inst. Fourier 59, 563–573 (2009)CrossRefzbMATHGoogle Scholar
  13. 13.
    Gálvez, J.A., Martínez, A.: Estimates in surfaces with positive constant Gauss Curvature. Proc. Am. Math. Soc. 128, 3655–3660 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    García-Martínez, S.C., Impera, D.: Height estimates and half-space theorems for spacelike hypersurfaces in generalized Robertson–Walker spacetimes. Differ. Geom. Appl. 32, 46–67 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    García-Martínez, S.C., Impera, D., Rigoli, M.: A sharp height estimates for compact hypersurfaces with constant \(k\)-mean curvature in warped product spaces. Proc. Edinb. Math. Soc. 58, 403–419 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gromov, M.: Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13, 178–215 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Heinz, E.: On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectificable boundary. Arch. Ration. Mech. Anal. 35, 249–252 (1969)CrossRefzbMATHGoogle Scholar
  18. 18.
    Hoffman, D., de Lira, J.H., Rosenberg, H.: Constant mean curvature surfaces in \(M^2\times \mathbb{R}\). Trans. Am. Math. Soc. 358, 491–507 (2006)CrossRefzbMATHGoogle Scholar
  19. 19.
    Latorre, J.M., Romero, A.: Uniqueness of noncompact spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes. Geom. Dedic. 93, 1–10 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lichnerowicz, A.: Variétés Riemanniennes à tenseur C non négatif. C. R. Acad. Sci. Paris Sér. A-B 271, A650–A653 (1970)zbMATHGoogle Scholar
  21. 21.
    Lichnerowicz, A.: Variétés Kählériennes à première classe de Chern non negative et variétés Riemanniennes à courbure de Ricci généralisée non negative. J. Differ. Geom. 6, 47–94 (1971)CrossRefzbMATHGoogle Scholar
  22. 22.
    López, R.: Area monotonicity for spacelike surfaces with constant mean curvature. J. Geom. Phys. 52, 353–363 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Manzano, J.M.: Estimates for constant mean curvature graphs in \(M\times {\mathbb{R}}\). Rev. Mat. Iberoam. 29, 1263–1281 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Morgan, F.: Manifolds with density. Not. Am. Math. Soc. 52, 853–858 (2005)MathSciNetzbMATHGoogle Scholar
  25. 25.
    O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London (1983)zbMATHGoogle Scholar
  26. 26.
    Sachs, R.K., Wu, H.: General Relativity for Mathematicians, Graduate Texts in Mathematics. Springer, New York (1977)CrossRefGoogle Scholar
  27. 27.
    Wei, G., Willie, W.: Comparison Geometry for the Bakry–Emery–Ricci Tensor. J. Differ. Geom. 83, 377–405 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2016

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de Campina GrandeCampina GrandeBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal da ParaíbaJoão PessoaBrazil

Personalised recommendations