Advertisement

Mathematische Annalen

, Volume 361, Issue 3–4, pp 927–941 | Cite as

Constrained matrix Li-Yau-Hamilton estimates on Kähler manifolds

  • Xin-An RenEmail author
  • Sha Yao
  • Li-Ju Shen
  • Guang-Ying Zhang
Article
  • 212 Downloads

Abstract

We derive a family of constrained matrix Li-Yau-Hamilton estimates on Kähler manifolds. As a result, we first get a constrained matrix Li-Yau-Hamilton estimate for heat equation on a Kähler manifold with fixed Kähler metric. Secondly, we get such estimate for forward conjugate heat equation on Kähler manifolds with time dependent Kähler metrics evolving under the Kähler-Ricci flow.

Mathematics Subject Classification

53C44 53C55 

Notes

Acknowledgments

The first named author would like to thank Professors Q.-K. Lu, S.-K. Wang and K. Wu for valuable discussions and comments. He would also like to thank Professor H.-W. Xu for helpful suggestions. This work is partially supported by the Fundamental Research Funds for Central Universities (Grant No. 2012QNA40).

References

  1. 1.
    Andrews, B.: Harnack inequalities for evolving hypersurfaces. Math. Z. 217, 179–197 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bando, S.: On classification of three-dimensional compact Kähler manifolds of nonnegative bisectional curvature. J. Differ. Geom. 19, 283–297 (1984)Google Scholar
  3. 3.
    Cao, H.D.: On Harnack’s inequalities for the Kähler-Ricci flow. Invent. Math. 109, 247–263 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Cao, H.D., Ni, L.: Matrix Li-Yau-Hamilton estimates for the heat equation on Kähler manifolds. Math. Ann. 331, 795–807 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chow, B.: On Harnack inequality and entropy for the Gaussian curvature flow. Commun. Pure Appl. Math. 44, 469–483 (1991)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chow, B.: The Yamabe flow on locally conformal flat manifolds with positive Ricci curvature. Commun. Pure Appl. Math. 45, 1003–1014 (1992)CrossRefzbMATHGoogle Scholar
  7. 7.
    Chow, B.: Interpolating between Li-Yau’s and Hamilton’s Harnack inequalities on surface. J. Partial Differ. Equ. 11, 137–140 (1998)zbMATHGoogle Scholar
  8. 8.
    Chow, B., Chu, S.C.: A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow. Math. Res. Lett. 2, 701–718 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Chow, B., Chu, S.C.: A geometric approach to the linear trace Harnack inequality for the Ricci flow. Math. Res. Lett. 3, 549–568 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chow, B., Hamilton, R.: Constrained and linear Harnack inequalities for parabolic equations. Invent. Math. 129, 213–238 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Chow, B., Knopf, D.: New Li-Yau-Hamilton inequalities for the Ricci flow via the space-time approach. J. Differ. Geom. 60, 1–54 (2002)Google Scholar
  12. 12.
    Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)Google Scholar
  13. 13.
    Hamilton, R.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24, 153–179 (1986)Google Scholar
  14. 14.
    Hamilton, R.: The Ricci flow on surfaces. Contemp. Math. 71, 237–262 (1993)CrossRefGoogle Scholar
  15. 15.
    Hamilton, R.: A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1, 113–126 (1993)zbMATHGoogle Scholar
  16. 16.
    Hamilton, R.: The Harnack estimate for the Ricci flow. J. Differ. Geom. 37, 225–243 (1993)zbMATHGoogle Scholar
  17. 17.
    Hamilton, R.: Monotonicity formulas for parabolic flows on manifolds. Commun. Anal. Geom. 1, 127–137 (1993)zbMATHGoogle Scholar
  18. 18.
    Hamilton, R.: The Harnack estimate for the mean curvature flow. J. Differ. Geom. 41, 215–226 (1995)zbMATHGoogle Scholar
  19. 19.
    Li, P., Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Mok, N.: The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature. J. Differ. Geom. 27, 179–214 (1988)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Ni, L.: A matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow. J. Differ. Geom. 75, 303–358 (2007)zbMATHGoogle Scholar
  22. 22.
    Ni, L.: Monotonicity and Kähler-Ricci flow. Contemp. Math. 367, 149–165 (2005)CrossRefGoogle Scholar
  23. 23.
    Ni, L.: A monotonicity formula on complete Kähler manifold with nonnegative bisectional curvature. J. Am. Math. Soc. 17, 909–946 (2004)CrossRefzbMATHGoogle Scholar
  24. 24.
    Ni, L., Tam, L.-F.: Plurisubharmonic functions and the Kähler-Ricci flow. Am. J. Math. 125, 623–654 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Ni, L., Tam, L.-F.: Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature. J. Differ. Geom. 64, 457–524 (2003)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Xin-An Ren
    • 1
    Email author
  • Sha Yao
    • 1
  • Li-Ju Shen
    • 1
  • Guang-Ying Zhang
    • 1
  1. 1.Department of MathematicsChina University of Mining and TechnologyXuzhouChina

Personalised recommendations