Advertisement

Journal of Geometric Analysis

, Volume 22, Issue 3, pp 763–779 | Cite as

Myers-Type Theorems and Some Related Oscillation Results

  • Paolo Mastrolia
  • Michele Rimoldi
  • Giona Veronelli
Article

Abstract

In this paper we study the behavior of solutions of a second-order differential equation. The existence of a zero and its localization allow us to get some compactness results. In particular we obtain a Myers-type theorem even in the presence of an amount of negative curvature. The technique we use also applies to the study of spectral properties of Schrödinger operators on complete manifolds.

Keywords

Myers-type theorems Oscillation Positioning of zeros 

Mathematics Subject Classification (2000)

53C20 34C10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambrose, W.: A theorem of Myers. Duke Math. J. 24, 345–348 (1957) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bianchini, B., Mari, L., Rigoli, M.: Spectral radius, index estimates for Schrödinger operators and geometric applications. J. Funct. Anal. 256(6), 1769–1820 (2009) (English summary) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Calabi, E.: On Ricci curvature and geodesics. Duke Math. J. 34, 667–676 (1967) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Fernández-López, M., García-Río, E.: A remark on compact Ricci solitons. Math. Ann. 340, 893–896 (2008) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Fischer-Colbrie, D.: On complete minimal surfaces with finite Morse index in three-manifolds. Invent. Math. 82(1), 121–132 (1985) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Galloway, G.J.: A generalization of Myers’ theorem and an application to relativistic cosmology. J. Differ. Geom. 14(1), 105–116 (1979) MathSciNetMATHGoogle Scholar
  7. 7.
    Galloway, G.J.: Compactness criteria for Riemannian manifolds. Proc. Am. Math. Soc. 84(1), 106–110 (1982) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Hille, E.: Non-oscillation theorems. Trans. Am. Math. Soc. 64, 234–252 (1948) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Leighton, W.: The detection of the oscillation of solutions of a second-order linear differential equation. Duke Math. J. 17, 57–61 (1950) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Moore, R.A.: The behavior of solutions of a linear differential equation of second-order. Pac. J. Math. 5, 125–145 (1955) MATHGoogle Scholar
  11. 11.
    Morgan, F.: Myers’ theorem with density. Kodai Math. J. 29(3), 455–461 (2006) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Morse, M.: The Calculus of Variations in the Large. American Mathematical Society Colloquium Publications, vol. 18. American Mathematical Society, Providence (1996). 368 pp. Reprint of the 1932 original Google Scholar
  13. 13.
    Myers, S.B.: Riemannian manifolds with positive mean curvature. Duke Math. J. 8, 401–404 (1941) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nehari, Z.: Oscillation criteria for second-order linear differential equations. Trans. Am. Math. Soc. 85, 428–445 (1957) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Pigola, S., Rigoli, M., Setti, A.G.: Vanishing and Finiteness Results in Geometric Analysis. A Generalization of the Bochner Technique. Progress in Mathematics, vol. 266. Birkhäuser, Basel (2008). 282 pp. MATHGoogle Scholar
  16. 16.
    Pigola, S., Rigoli, M., Rimoldi, M., Setti, A.G.: Ricci almost solitons. Ann. Sc. Norm. Sup. Pisa (2010, to appear). arXiv:1003.2945v1
  17. 17.
    Qian, Z.: Estimates for the weighted volumes and applications. Q. J. Math. 48(190), 235–242 (1997) MATHCrossRefGoogle Scholar
  18. 18.
    Rimoldi, M.: A remark on Einstein warped products. Pac. J. Math. (2010, to appear). arXiv:1004.3866v3
  19. 19.
    Swanson, C.A.: Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, San Diego (1968) MATHGoogle Scholar
  20. 20.
    Wei, G., Wylie, W.: Comparison geometry for the Bakry–Emery Ricci tensor. J. Differ. Geom. 83, 377–405 (2009) MathSciNetMATHGoogle Scholar
  21. 21.
    Wylie, W.: Complete shrinking Ricci solitons have finite fundamental group. Proc. Am. Math. Soc. 136(5), 1803–1806 (2008) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2011

Authors and Affiliations

  • Paolo Mastrolia
    • 1
  • Michele Rimoldi
    • 1
  • Giona Veronelli
    • 1
  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanoItaly

Personalised recommendations