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An improvement of the Myers theorem

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Abstract

Let (Mg) be a complete Riemannian manifold of dimension n. We consider a Ricci curvature condition on M to prove a compactness theorem including a diameter estimate. It is an improvement of the classical theorem of Myers, and it is comparable with some other Myers type compactness theorems.

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References

  1. Calabi, E.: On Ricci curvature and geodesics. Duke Math. J. 34, 667–676 (1967)

    Article  MathSciNet  Google Scholar 

  2. Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17, 15–53 (1982)

    Article  MathSciNet  Google Scholar 

  3. Elworthy, K.D., Rosenberg, S.: Manifolds with wells of negative curvature. Invent. Math. 103, 471–495 (1991)

    Article  MathSciNet  Google Scholar 

  4. Hartman, P.: Ordinary Differential Equations, 2nd edn. SIAM, Philadelphia (2002)

    Book  Google Scholar 

  5. Kelley, W.G., Peterson, A.C.: The Theory of Differential Equations Classical and Qualitative. Springer, New York (2010)

    Book  Google Scholar 

  6. Li, X.-M.: On extensions of Myers’ Theorem. Bull. London Math. Soc. 27, 392–396 (1995)

    Article  MathSciNet  Google Scholar 

  7. Myers, S.B.: Riemannian manifolds with positive mean curvature. Duke Math. J. 8, 401–404 (1941)

    Article  MathSciNet  Google Scholar 

  8. Pigola, S., Rigoli, M., Setti, A.G.: Vanishing and Finiteness Results in Geometric Analysis. Birkhauser, Boston (2008)

    Google Scholar 

  9. Tipler, F.J.: General relativity and conjugate ordinary differential equations. J. Differ. Equ. 30, 165–174 (1978)

    Article  MathSciNet  Google Scholar 

  10. Wan, J.: An extension of Bonnet-Myers theorem. Math. Z. 291, 195–197 (2019)

    Article  MathSciNet  Google Scholar 

  11. Willett, D.: Classification of second order linear differential equations with respect to oscillation. Adv. Math. 3, 594–623 (1969)

    Article  MathSciNet  Google Scholar 

  12. Wu, J.-Y.: Complete manifolds with a little negative curvature. Am. J. Math. 113, 567–572 (1991)

    Article  MathSciNet  Google Scholar 

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Correspondence to Murat Limoncu.

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Limoncu, M. An improvement of the Myers theorem. manuscripta math. 174, 991–1004 (2024). https://doi.org/10.1007/s00229-024-01534-6

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  • DOI: https://doi.org/10.1007/s00229-024-01534-6

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