Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

RIEMANNIAN AND LORENTZIAN VOLUME COMPARISONS WITH THE BAKRY-EMERY RICCI TENSOR

  • Kim, Jong Ryul (Department of Mathematics Kunsan National University)
  • Received : 2021.09.15
  • Accepted : 2021.11.25
  • Published : 2021.11.15
Download PDF
⟨ Previous Next ⟩

Abstract

The Bishop and Bishop-Gromov volume comparisons with the Bakry-Emery Ricci tensor in a metric measure space are studied by the comparisons of the Jacobi differential equations in a Riemannian and Lorentzian manifold.

Keywords

References

  1. D. Bakry, M. Emery, Diffusions hypercontractive, Seminaire de Probabilites XIX. Lecture Notes Math. 1123, 117-206 (1985)
  2. Jeffrey S. Case, Singularity theorems and the Lorentzian splitting theorem for the Bakry-Emery-Ricci tensor, J. Geom. Phys., 60 (2010), no. 3, 477-490. https://doi.org/10.1016/j.geomphys.2009.11.001
  3. P.E. Ehrlich, Y.-T. Jung, S.-B. Kim, Volume comparison theorems for Lorentzian manifolds, Geom. Dedicata 73 (1998), no. 1, 39-56. https://doi.org/10.1023/A:1005096913126
  4. J.-H. Eschenburg, Comparison theorems and hypersurfaces, Manuscripta Math. 59 (1987), 295-323. https://doi.org/10.1007/BF01174796
  5. J.-H. Eschenburg, J. O'Sullivan, Jacobi tensors and Ricci curvature, Math. Ann., 252 (1980), 1-26. https://doi.org/10.1007/BF01420210
  6. J.R. Kim, Comparisons of the Lorentzian volumes between level hypersurfaces, J. Geom. Phys., 59 (2009), no. 7, 1073-1078. https://doi.org/10.1016/j.geomphys.2009.04.011
  7. J.R. Kim, Relative Lorentzian volume comparison with integral Ricci and scalar curvature bound, J. Geom. Phys., 61 (2011), no. 6, 1061-1069. https://doi.org/10.1016/j.geomphys.2011.02.005
  8. S.-H. Paeng, Volume comparison between spacelike hypersurfaces in a Lorentzian manifold with integral Ricci curvature bounds, Gen. Relativ. Gravit., 43 (2011), 2089-2102. https://doi.org/10.1007/s10714-010-1106-1
  9. P. Petersen, G. Wei, Relative volume comparison with integral curvature bounds, Geom. Funct. Anal., 7 (1997), 1031-1045. https://doi.org/10.1007/s000390050036
  10. Qi-hua Ruan, Two rigidity theorems on manifolds with Bakry-Emery Ricci curvature, Proc. Japan Acad. Ser. A, 85 (2009), no. 6, 71-74.
  11. G. Wei, W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differ. Geom., 83 (2009), no. 2, 377-405.
  12. Jia-Yong Wu, Upper Bounds on the First Eigenvalue for a Diffusion Operator via Bakry Emery Ricci Curvature II, Results Math. 63 (2013), no. 3-4, 1079-1094. https://doi.org/10.1007/s00025-012-0254-x
  13. J.-G. Yun, Volume comparison for Lorentzian warped products with integral curvature bounds, J. Geom. Phys., 57 (2007), no. 3, 903-912. https://doi.org/10.1016/j.geomphys.2006.07.001