The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 783–819 | Cite as

Linear Trace Li–Yau–Hamilton Inequality for the CR Lichnerowicz–Laplacian Heat Equation

  • Shu-Cheng Chang
  • Ting-Hui ChangEmail author
  • Yen-Wen Fan


In this paper, we study the CR Lichnerowicz–Laplacian heat equation deformation of (1,1)-tensors on a complete strictly pseudoconvex CR (2n+1)-manifold. We derive the linear trace version of the Li–Yau–Hamilton inequality for positive solutions of the CR Lichnerowicz–Laplacian heat equation. We also obtain a nonlinear version of the Li–Yau–Hamilton inequality for the CR Lichnerowicz–Laplacian heat equation coupled with the CR Yamabe flow and trace Harnack inequality for the CR Yamabe flow.


Li–Yau–Hamilton inequality Bochner–Weitzenböck formula CR Hodge–Laplacian CR Lichnerowicz–Laplacian heat equation CR Yamabe flow Torsion flow 

Mathematics Subject Classification

32V05 32V20 53C56 


  1. 1.
    Chow, B.: The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Commun. Pure Appl. Math. XLV, 1003–1014 (1992) CrossRefGoogle Scholar
  2. 2.
    Cao, H.-D.: On Harnack inequalities for the Kähler–Ricci flow. Invent. Math. 109, 247–263 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Chang, S.-C., Cheng, J.-H.: The Harnack estimate for the Yamabe flow on CR manifolds of dimension 3. Ann. Glob. Anal. Geom. 21, 111–121 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Chang, D.-C., Chang, S.-C., Tie, J.-Z.: Calabi-Yau theorem and Hodge–Laplacian heat equation in a closed strictly pseudoconvex CR manifold. Preprint (2013) Google Scholar
  5. 5.
    Chow, B., Hamilton, R.S.: Constrained and linear Harnack inequalities for parabolic equations. Invent. Math. 129, 213–238 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chang, S.-C., Kuo, T.-J., Lai, S.-H.: Li–Yau gradient estimate and entropy formulae for the CR heat equation in a closed pseudohermitian 3-manifold. J. Differ. Geom. 89, 185–216 (2011) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Chang, S.-C., van Koert, O., Wu, C.-T.: The torsion flow in a closed pseudohermitian 3-manifold. Preprint (2013) Google Scholar
  8. 8.
    Hamilton, R.S.: The Harnack estimate for the Ricci flow. J. Differ. Geom. 37(1), 225–243 (1993) zbMATHGoogle Scholar
  9. 9.
    Hamilton, R.S.: Harnack estimate for the mean curvature flow. J. Differ. Geom. 41(1), 215–226 (1995) zbMATHGoogle Scholar
  10. 10.
    Karp, L., Li, P.: The heat equation on complete Riemannian manifolds. Unpublished Google Scholar
  11. 11.
    Lee, J.M.: Pseudo-Einstein structure on CR manifolds. Am. J. Math. 110, 157–178 (1988) CrossRefzbMATHGoogle Scholar
  12. 12.
    Lee, J.M.: The Fefferman metric and pseudohermitian invariants. Trans. Am. Math. Soc. 296, 411–429 (1986) zbMATHGoogle Scholar
  13. 13.
    Li, P., Tam, L.-F.: The heat equation and harmonic maps of complete manifolds. Invent. Math. 105, 1–46 (1991) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Li, P., Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Morrey, C.: Multiple Integrals in Calculus of Variations. Springer, New York (1966) zbMATHGoogle Scholar
  16. 16.
    Ni, L.: A Monotonicity Formula on Complete Kähler Manifolds with Nonnegative Bisectional Curvature. J. Am. Math. Soc. 17, 909–946 (2004) CrossRefzbMATHGoogle Scholar
  17. 17.
    Ni, L.: An optimal gap theorem. Invent. Math. 189, 737–761 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ni, L., Tam, L.-F.: Plurisubharmonic functions and the Kähler-Ricci flow. Am. J. Math. 125, 623–654 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Ni, L., Tam, L.-F.: Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature. J. Differ. Geom. 64(3), 457–524 (2003) zbMATHMathSciNetGoogle Scholar
  20. 20.
    Ni, L., Tam, L.-F.: Kähler–Ricci flow and Poincare-Lelong equation. Commun. Anal. Geom. 12(1), 111–114 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Ni, L., Niu, Y.-Y.: Sharp differential estimates of Li–Yau–Hamilton type for positive (p,p)-forms on Kähler manifolds. Commun. Pure Appl. Math. 64, 920–974 (2011) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Taida Institute for Mathematical Sciences (TIMS)National Taiwan UniversityTaipeiTaiwan
  2. 2.Institute of MathematicsAcademia SinicaTaipeiTaiwan
  3. 3.Department of MathematicsNational Taiwan UniversityTaipeiTaiwan

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