Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EVOLUTION AND MONOTONICITY FOR A CLASS OF QUANTITIES ALONG THE RICCI-BOURGUIGNON FLOW

  • Daneshvar, Farzad (Department of Mathematics and Computer Science Mahani Mathematical Research Center Shahid Bahonar University of Kerman) ;
  • Razavi, Asadollah (Department of Mathematics and Computer Science Mahani Mathematical Research Center Shahid Bahonar University of Kerman)
  • Received : 2018.08.02
  • Accepted : 2019.07.25
  • Published : 2019.11.01
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Abstract

In this paper we consider the monotonicity of the lowest constant ${\lambda}_a^b(g)$ under the Ricci-Bourguignon flow and the normalized Ricci-Bourguignon flow such that the equation $$-{\Delta}u+au\;{\log}\;u+bRu={\lambda}_a^b(g)u$$ with ${\int}_{M}u^2dV=1$, has positive solutions, where a and b are two real constants. We also construct various monotonic quantities under the Ricci-Bourguignon flow and the normalized Ricci-Bourguignon flow. Moreover, we prove that a compact steady breather which evolves under the Ricci-Bourguignon flow should be Ricci-flat.

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References

  1. X. Cao, Eigenvalues of ($--\Delta}+\frac-R}-2}$) on manifolds with nonnegative curvature operator, Math. Ann. 337 (2007), no. 2, 435-441. https://doi.org/10.1007/s00208-006-0043-5
  2. X. Cao, First eigenvalues of geometric operators under the Ricci flow, Proc. Amer. Math. Soc. 136 (2008), no. 11, 4075-4078. https://doi.org/10.1090/S0002-9939-08-09533-6
  3. B. Chen, Q. He, and F. Zeng, Monotonicity of eigenvalues of geometric operators along the Ricci-Bourguignon flow, Pacific J. Math. 296 (2018), no. 1, 1-20. https://doi.org/10.2140/pjm.2018.296.1
  4. B. Chow et al., The Ricci flow: techniques and applications. Part I, Mathematical Surveys and Monographs, 135, American Mathematical Society, Providence, RI, 2007.
  5. L. Cremaschi, Some variations on Ricci flow, Ph.D Thesis, 2016.
  6. H. Guo, R. Philipowski, and A. Thalmaier, Entropy and lowest eigenvalue on evolving manifolds, Pacific J. Math. 264 (2013), no. 1, 61-81. https://doi.org/10.2140/pjm.2013.264.61
  7. G. Huang and Z. Li, Evolution of a geometric constant along the Ricci flow, J. Inequal. Appl. 2016, Paper No. 53, 11 pp. https://doi.org/10.1186/s13660-016-1003-6
  8. J.-F. Li, Eigenvalues and energy functionals with monotonicity formulae under Ricci flow, Math. Ann. 338 (2007), no. 4, 927-946. https://doi.org/10.1007/s00208-007-0098-y
  9. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, (2002) arXiv:math/0211159.
  10. P. Topping, Lectures on the Ricci Flow, London Mathematical Society Lecture Note Series, 325, Cambridge University Press, Cambridge, 2006. https://doi.org/10.1017/CBO9780511721465
  11. L. F. Wang, Monotonicity of eigenvalues and functionals along the Ricci-Bourguignon flow, J. Geom. Anal. 29 (2019), no. 2, 1116-1135. https://doi.org/10.1007/s12220-018-0030-6