Geometriae Dedicata

, Volume 169, Issue 1, pp 225–237 | Cite as

Some triviality results for quasi-Einstein manifolds and Einstein warped products

Original Paper


In this paper we prove a number of triviality results for Einstein warped products and quasi-Einstein manifolds using different techniques and under assumptions of various nature. In particular we obtain and exploit gradient estimates for solutions of weighted Poisson-type equations and adaptations to the weighted setting of some Liouville-type theorems.


Einstein warped products Quasi-Einstein manifolds  Triviality  Gradient estimates 

Mathematics Subject Classification (2000)




We wish to thank Stefano Pigola and Jeffrey Case for valuable suggestions and useful comments on earlier versions of the paper. We thank the anonymous referee for useful suggestions that helped to improve the exposition. The first author would also like to thank Francesca Savini for some useful remarks.


  1. 1.
    Anderson, M.T.: Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds. Geom. Funct. Anal. 9(2), 855–967 (1999)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Besse, A.: Einstein Manifolds. Reprint of the 1987 edition. Classics in Mathematics. Springer, Berlin, p. xii+516 (2008)Google Scholar
  3. 3.
    Calabi, E.: An extension of Hopf’s maximum principle with an application to Riemannian geometry. Duke Math. J. 25, 45–56 (1957)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Case, J.S., Shu, Y.-J., Wei, G.: Rigidity of quasi-Einstein metrics. Diff. Geom. Appl. 29(1), 93–100 (2011)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Case, J.S.: On the nonexistence of quasi-Einstein metrics. Pac. J. Math. 248(2), 277–284 (2010)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Case, J.S.: Smooth metric measure spaces and quasi-Einstein metrics. Intern. J. Math. 23, 1250110 (2012)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Corvino, J.: Scalar curvature deformation and gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214(1), 137–189 (2000)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Eminenti, M., La Nave, G., Mantegazza, C.: Ricci solitons: the equation point of view. Manuscr. Math. 127, 345–367 (2008)CrossRefMATHGoogle Scholar
  9. 9.
    Hamilton R.S.: The Ricci flow on surfaces. Mathematics and general relativity (Santa Cruz, CA, 1986). Contemp. Math., Am. Math. Soc. 71, 237–262 (1988)Google Scholar
  10. 10.
    He, C., Petersen, P., Wylie, W.: On the classification of warped product Einstein metrics. Commun. Anal. Geom. 20(2), 271–311 (2012)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Kim, D.-S., Kim, Y.H.: Compact Einstein warped product spaces with nonpositive scalar curvature. Proc. Amer. Math. Soc. 131, 2573–2576 (2003)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Li, X.-D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl. 84, 1295–1361 (2005)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Lu, H., Page, D.N., Pope, C.N.: New inhomogeneous Einstein metrics on sphere bundles over Einstein-Kaehler manifolds. Phys. Lett. B 593, 218–226 (2004)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Mari, L., Rigoli, M., Setti, A.G.: Keller-Osserman conditions for diffusion-type operators on Riemannian manifolds. J. Funct. Anal. 258, 665–712 (2010)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Mastrolia, P., Rigoli, M.: Diffusion-type operators, Liouville theorems and gradient estimates on complete manifolds. Nonlinear Anal. 72, 3767–3785 (2010)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Perelman G.: Ricci flow with surgery on three manifolds. arXiv:math/0303109v1 [math.DG] (2003)Google Scholar
  17. 17.
    Petersen, P., Wylie, W.: On gradient Ricci solitons with symmetry. Proc. Amer. Math. Soc. 137, 2085–2092 (2009)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Petersen, P., Wylie, W.: Rigidity of gradient Ricci solitons. Pac. J. Math. 241, 329–345 (2009)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Petersen, P., Wylie, W.: On the classification of gradient Ricci solitons. Geom. Topol. 14(4), 2277–2300 (2010)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Pigola, S., Rigoli, M., Rimoldi, M., Setti, A.G.: Ricci almost solitons. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) X(4), 757–799 (2011)Google Scholar
  21. 21.
    Pigola, S., Rigoli, M., Setti, A.G.: Volume growth, “a priori” estimates, and geometric applications. Geom. Funct. Anal. 13(6), 1302–1328 (2003)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Pigola, S., Rigoli, M., Setti, A.G.: A Liouville-type result for quasi-linear elliptic equations on complete Riemannian manifolds. J. Funct. Anal. 219(2), 400–432 (2005)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Pigola, S., Rigoli, M., Setti, A.G.: Maximum principles on Riemannian manifolds and applications. Mem. Amer. Math. Soc. 174, 1–99 (2005)MathSciNetGoogle Scholar
  24. 24.
    Pigola, S., Rimoldi, M., Setti, A.G.: Remarks on non-compact gradient Ricci solitons. Math. Z. 268(3), 777–790 (2011)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Qian, Z.: Estimates for weighted volumes and applications. Q. J. Math. Oxf. 48, 235–242 (1997)CrossRefMATHGoogle Scholar
  26. 26.
    Rimoldi, M.: A remark on Einstein warped product. Pac. J. Math. 252(1), 207–218 (2011)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Rimoldi, M.: Rigidity Results for Lichnerowicz Bakry-Emery Ricci Tensors. Ph. D. thesis. Universitá degli Studi di Milano. Avaiable at (2012)
  28. 28.
    Schoen, R., Yau, S.-T.: Lectures on differential geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology. International Press, Cambridge (1994)Google Scholar
  29. 29.
    Wei, G., Wylie, W.: Comparison geometry for the smooth metric measure spaces. In: Proceedings of the 4th International Congress of Chinese Mathematicians. II, pp. 191–202. Hangzhou, China (2007)Google Scholar
  30. 30.
    Wei, G., Wylie, W.: Comparison geometry for the Bakry-Emery Ricci tensor. J. Diff. Geom. 83(2), 377–405 (2009)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanItaly
  2. 2.Dipartimento di Scienza e Alta TecnologiaUniversità degli Studi dell’InsubriaComoItaly

Personalised recommendations