Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON EVOLUTION OF FINSLER RICCI SCALAR

  • Bidabad, Behroz (Faculty of Mathematics and Computer Science Amirkabir University of Technology (Tehran Polytechnic)) ;
  • Sedaghat, Maral Khadem (Faculty of Mathematics and Computer Science Amirkabir University of Technology (Tehran Polytechnic))
  • Received : 2017.06.30
  • Accepted : 2018.01.30
  • Published : 2018.05.01
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Abstract

Here, we calculate the evolution equation of the reduced hh-curvature and the Ricci scalar along the Finslerian Ricci flow. We prove that Finsler Ricci flow preserves positivity of the reduced hh-curvature on finite time. Next, it is shown that evolution of Ricci scalar is a parabolic-type equation and moreover if the initial Finsler metric is of positive flag curvature, then the flag curvature, as well as the Ricci scalar, remain positive as long as the solution exists. Finally, we present a lower bound for Ricci scalar along Ricci flow.

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Acknowledgement

Supported by : Iran National Science Foundation

References

  1. H. Akbar-Zadeh, Initiation to Global Finslerian Geometry, North-Holland Mathematical Library, 68, Elsevier Science B.V., Amsterdam, 2006.
  2. S. Azami and A. Razavi, Existence and uniqueness for solution of Ricci flow on Finsler manifolds, Int. J. Geom. Methods Mod. Phys. 10 (2013), no. 3, 1250091, 21 pp.
  3. D. Bao, On two curvature-driven problems in Riemann-Finsler geometry, in Finsler geometry, Sapporo 2005-in memory of Makoto Matsumoto, 19-71, Adv. Stud. Pure Math., 48, Math. Soc. Japan, Tokyo, 2007.
  4. D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics, 200, Springer-Verlag, New York, 2000.
  5. D. Bao and Z. M. Shen, On the volume of unit tangent spheres in a Finsler manifold, Results Math. 26 (1994), no. 1-2, 1-17. https://doi.org/10.1007/BF03322283
  6. B. Bidabad and M. K. Sedaghat, Hamilton's Ricci Flow on Finsler Spaces, arXiv:1508.02893, 2015.
  7. B. Bidabad and M. K. Sedaghat, Ricci flow on Finsler surfaces, J. Geom. Phys., 2018.
  8. B. Bidabad, A. Shahi, and M. Yarahmadi, Deformation of Cartan curvature on Finsler manifolds, Bull. Korean Math. Soc. 54, (2017), no. 6, 2119-2139. https://doi.org/10.4134/BKMS.B160780
  9. B. Bidabad and M. Yarahmadi, On quasi-Einstein Finsler spaces, Bull. Iranian Math. Soc. 40 (2014), no. 4, 921-930.
  10. S. Lakzian, Differential Harnack estimates for positive solutions to heat equation under Finsler-Ricci flow, Pacific J. Math. 278 (2015), no. 2, 447-462. https://doi.org/10.2140/pjm.2015.278.447
  11. B. Shen, Twisted Ricci flow on a class of Finsler metrics and its solitons, Differential Geom. Appl. 46 (2016), 132-145. https://doi.org/10.1016/j.difgeo.2016.02.005
  12. P. Topping, Lectures on the Ricci Flow, London Mathematical Society Lecture Note Series, 325, Cambridge University Press, Cambridge, 2006.
  13. S. I. Vacaru, Metric compatible or non-compatible Finsler-Ricci flows, Int. J. Geom. Methods Mod. Phys. 9 (2012), no. 5, 1250041, 26 pp.
  14. M. Yarahmadi and B. Bidabad, On compact Ricci solitons in Finsler geometry, C. R. Math. Acad. Sci. Paris 353 (2015), no. 11, 1023-1027. https://doi.org/10.1016/j.crma.2015.09.012
  15. M. Yarahmadi and B. Bidabad, Convergence of Finslerian metrics under Ricci flow, Sci. China Math. 59 (2016), no. 4, 741-750. https://doi.org/10.1007/s11425-015-5092-3