대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제34권3호
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  • Pages.863-879
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  • 2019
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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MYERS' THEOREM FOR COMPLETE RIEMANNIAN HYPERSURFACES

  • Fatemi, Seyed Hamed (Department of Pure Mathematics Faculty of Math. Sciences Tarbiat Modares University)
  • 투고 : 2018.04.22
  • 심사 : 2018.10.24
  • 발행 : 2019.07.31
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초록

In this paper we study the Myers' theorem for orientable Riemannian hypersurfaces under some restrictions on the mean curvature, the second-order mean curvature, or the divergence of the shape operator and give some estimates for the diameter of such hypersurfaces.

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참고문헌

  1. L. J. Alas, P. Mastrolia, and M. Rigoli, Maximum principles and geometric applications, Springer Monographs in Mathematics, Springer, Cham, 2016. https://doi.org/10.1007/978-3-319-2437-5
  2. M. P. Cavalcante, J. Q. Oliveira, and M. S. Santos, Compactness in weighted manifolds and applications, Results Math. 68 (2015), no. 1-2, 143-156. https://doi.org/10.1007/s00025-014-0427-x
  3. F. Fang, X.-D. Li, and Z. Zhang, Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Emery Ricci curvature, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 2, 563-573. https://doi.org/10.5802/aif.2440
  4. A. Futaki, H. Li, and X.-D. Li, On the rst eigenvalue of the Witten-Laplacian and the diameter of compact shrinking solitons, Ann. Global Anal. Geom. 44 (2013), no. 2, 105-114. https://doi.org/10.1007/s10455-012-9358-5
  5. X.-D. Li, Liouville theorems for symmetric diusion operators on complete Riemannian manifolds, J. Math. Pures Appl. (9) 84 (2005), no. 10, 1295-1361. https://doi.org/10.1016/j.matpur.2005.04.002
  6. M. Limoncu, The Bakry-Emery Ricci tensor and its applications to some compactness theorems, Math. Z. 271 (2012), no. 3-4, 715-722. https://doi.org/10.1007/s00209-011-0886-7
  7. O. Munteanu and J. Wang, Smooth metric measure spaces with non-negative curvature, Comm. Anal. Geom. 19 (2011), no. 3, 451-486. https://doi.org/10.4310/CAG.2011.v19.n3.a1
  8. S. B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401-404. http://projecteuclid.org/euclid.dmj/1077492655 https://doi.org/10.1215/S0012-7094-41-00832-3
  9. H. Tadano, Remark on a diameter bound for complete Riemannian manifolds with positive Bakry-Emery Ricci curvature, Diffierential Geom. Appl. 44 (2016), 136-143. https://doi.org/10.1016/j.difgeo.2015.11.001
  10. G. Wei and W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377-405. https://doi.org/10.4310/jdg/1261495336
  11. J.-Y. Wu, A note on the splitting theorem for the weighted measure, Ann. Global Anal. Geom. 43 (2013), no. 3, 287-298. https://doi.org/10.1007/s10455-012-9346-9
  12. J.-Y. Wu, Myers' type theorem with the Bakry-Emery Ricci tensor, Ann. Global Anal. Geom. 54 (2018), no. 4, 541-549. https://doi.org/10.1007/s10455-018-9613-5
  13. S. Zhang, A theorem of Ambrose for Bakry-Emery Ricci tensor, Ann. Global Anal. Geom. 45 (2014), no. 3, 233-238. https://doi.org/10.1007/s10455-013-9396-7