Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

HARNACK ESTIMATES FOR NONLINEAR BACKWARD HEAT EQUATIONS WITH POTENTIALS ALONG THE RICCI-BOURGUIGNON FLOW

  • Wang, Jian-Hong (School of Mathematical Sciences East China Normal University)
  • Received : 2019.01.16
  • Accepted : 2019.08.26
  • Published : 2020.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we derive various differential Harnack estimates for positive solutions to the nonlinear backward heat type equations on closed manifolds coupled with the Ricci-Bourguignon flow, which was done for the Ricci flow by J.-Y. Wu [30]. The proof follows exactly the one given by X.-D. Cao [4] for the linear backward heat type equations coupled with the Ricci flow.

Keywords

References

  1. B. Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z. 217 (1994), no. 2, 179-197. https://doi.org/10.1007/BF02571941
  2. J.-P. Bourguignon, Ricci curvature and Einstein metrics, in Global differential geometry and global analysis (Berlin, 1979), 42-63, Lecture Notes in Math., 838, Springer, Berlin, 1981.
  3. H.-D. Cao, On Harnack's inequalities for the Kahler-Ricci flow, Invent. Math. 109 (1992), no. 2, 247-263. https://doi.org/10.1007/BF01232027
  4. X. Cao, Differential Harnack estimates for backward heat equations with potentials under the Ricci flow, J. Funct. Anal. 255 (2008), no. 4, 1024-1038. https://doi.org/10.1016/j.jfa.2008.05.009
  5. X. Cao, H. Guo, and H. Tran, Harnack estimates for conjugate heat kernel on evolving manifolds, Math. Z. 281 (2015), no. 1-2, 201-214. https://doi.org/10.1007/s00209-015-1479-7
  6. X. Cao and R. S. Hamilton, Differential Harnack estimates for time-dependent heat equations with potentials, Geom. Funct. Anal. 19 (2009), no. 4, 989-1000. https://doi.org/10.1007/s00039-009-0024-4
  7. X. Cao and Z. Zhang, Differential Harnack estimates for parabolic equations, in Complex and differential geometry, 87-98, Springer Proc. Math., 8, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-20300-8_5
  8. G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza, and L. Mazzieri, The Ricci-Bourguignon flow, Pacific J. Math. 287 (2017), no. 2, 337-370. https://doi.org/10.2140/pjm.2017.287.337
  9. H.-B. Cheng, A new Li-Yau-Hamilton estimate for the Ricci flow, Comm. Anal. Geom. 14 (2006), no. 3, 551-564. http://projecteuclid.org/euclid.cag/1175790091 https://doi.org/10.4310/CAG.2006.v14.n3.a5
  10. B. Chow, The Ricci flow on the 2-sphere, J. Differential Geom. 33 (1991), no. 2, 325-334. http://projecteuclid.org/euclid.jdg/1214446319
  11. B. Chow, On Harnack's inequality and entropy for the Gaussian curvature flow, Comm. Pure Appl. Math. 44 (1991), no. 4, 469-483. https://doi.org/10.1002/cpa.3160440405
  12. B. Chow, The Yamabe flow on locally conformally at manifolds with positive Ricci curvature, Comm. Pure Appl. Math. 45 (1992), no. 8, 1003-1014. https://doi.org/10.1002/cpa.3160450805
  13. B. Chow and R. S. Hamilton, Constrained and linear Harnack inequalities for parabolic equations, Invent. Math. 129 (1997), no. 2, 213-238. https://doi.org/10.1007/s002220050162
  14. B. Chow and D. Knopf, New Li-Yau-Hamilton inequalities for the Ricci flow via the space-time approach, J. Differential Geom. 60 (2002), no. 1, 1-54. http://projecteuclid.org/euclid.jdg/1090351083
  15. A. E. Fischer, An introduction to conformal Ricci flow, Classical Quantum Gravity 21 (2004), no. 3, S171-S218. https://doi.org/10.1088/0264-9381/21/3/011
  16. C. M. Guenther, The fundamental solution on manifolds with time-dependent metrics, J. Geom. Anal. 12 (2002), no. 3, 425-436. https://doi.org/10.1007/BF02922048
  17. H. Guo and M. Ishida, Harnack estimates for nonlinear backward heat equations in geometric flows, J. Funct. Anal. 267 (2014), no. 8, 2638-2662. https://doi.org/10.1016/j.jfa.2014.08.006
  18. H. Guo and M. Ishida, Harnack estimates for nonlinear heat equations with potentials in geometric flows, Manuscripta Math. 148 (2015), no. 3-4, 471-484. https://doi.org/10.1007/s00229-015-0757-3
  19. R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988. https://doi.org/10.1090/conm/071/954419
  20. R. S. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993), no. 1, 225-243. http://projecteuclid.org/euclid.jdg/1214453430 https://doi.org/10.4310/jdg/1214453430
  21. R. S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993), no. 1, 113-126. https://doi.org/10.4310/CAG.1993.v1.n1.a6
  22. R. S. Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1995), no. 1, 215-226. http://projecteuclid.org/euclid.jdg/1214456010 https://doi.org/10.4310/jdg/1214456010
  23. S. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal. 255 (2008), no. 4, 1008-1023. https://doi.org/10.1016/j.jfa.2008.05.014
  24. P. Li and S.-T. Yau, On the parabolic kernel of the Schrodinger operator, Acta Math. 156 (1986), no. 3-4, 153-201. https://projecteuclid.org/euclid.acta/1485890415 https://doi.org/10.1007/BF02399203
  25. L. Ni, A note on Perelman's LYH-type inequality, Comm. Anal. Geom. 14 (2006), no. 5, 883-905. http://projecteuclid.org/euclid.cag/1175790872 https://doi.org/10.4310/CAG.2006.v14.n5.a3
  26. L. Ni, A matrix Li-Yau-Hamilton estimate for Kahler-Ricci flow, J. Differential Geom. 75 (2007), no. 2, 303-358.
  27. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, 2002. arXiv math.DG/0211159v1.
  28. N. Sesum, G. Tian, and X.-D. Wang, Notes on Perelman's paper on the entropy formula for the Ricci flow and its geometric applications, preprint, 2003.
  29. J.-Y. Wu, Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds, J. Math. Anal. Appl. 369 (2010), no. 1, 400-407. https://doi.org/10.1016/j.jmaa.2010.03.055
  30. J.-Y. Wu, Differential Harnack inequalities for nonlinear heat equations with potentials under the Ricci flow, Pacific J. Math. 257 (2012), no. 1, 199-218. https://doi.org/10.2140/pjm.2012.257.199
  31. J.-Y. Wu and Y. Zheng, Interpolating between constrained Li-Yau and Chow-Hamilton Harnack inequalities on a surface, Arch. Math. (Basel) 94 (2010), no. 6, 591-600. https://doi.org/10.1007/s00013-010-0135-z
  32. Y. Yang, Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds, Proc. Amer. Math. Soc. 136 (2008), no. 11, 4095-4102. https://doi.org/10.1090/S0002-9939-08-09398-2
  33. S.-T. Yau, On the Harnack inequalities of partial differential equations, Comm. Anal. Geom. 2 (1994), no. 3, 431-450. https://doi.org/10.4310/CAG.1994.v2.n3.a3
  34. S.-T. Yau, Harnack inequality for non-self-adjoint evolution equations, Math. Res. Lett. 2 (1995), no. 4, 387-399. https://doi.org/10.4310/MRL.1995.v2.n4.a2
  35. R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom. 39 (1994), no. 1, 35-50. http://projecteuclid.org/euclid.jdg/1214454674
  36. Q. S. Zhang, Some gradient estimates for the heat equation on domains and for an equation by Perelman, Int. Math. Res. Not. 2006 (2006), Art. ID 92314, 39 pp. https://doi.org/10.1155/IMRN/2006/92314