My Research

Research interests :
In my research, I have been concentrating on the study of geometric flows and their application to differential geometry and
topology. My main research topics focus on the study of singularities, long time existence and convergence of the Ricci
flow and other geometric flows.

Publications

  1. Isoperimetric Estimate for the Ricci Flow on S^2 x S^1
    Comm. Anal. Geom., Vol. 13(4): 727-740, 2005.
    abstract
    , pdf


  2. Eigenvalues of (-\Delta +R/2) on Manifolds with Nonnegative Curvature Operator
    Mathematische Annalen, Vol. 337 (2): 435-441, 2007.
    abstract, pdf


  3. Compact Gradient Shrinking Ricci Solitons with Positive Curvature Operator
    Journal of Geometric Analysis, Vol. 17 (3): 451-459, 2007.
    abstract, pdf


  4. Dimension Reduction under the Ricci Flow on Manifolds with Positive Curvature Operator
    Pacific J. Math., Vol. 232 (2): 263-268, 2007.
    abstract, pdf


  5. Cross Curvature Flow on Locally Homogeneous Three-manifolds (I)
    (with Yilong Ni and Laurent Saloff-Coste) Pacific J. Math., Vol. 236 (2): 263-281, 2008.
    abstract, pdf


  6. First Eigenvalues of Geometric Operators under the Ricci Flow
    Proc. Amer. Math. Soc., Vol. 136 (11): 4075-4078, 2008.
    abstract, pdf


  7. Differential Harnack Estimates for Backward Heat Equations with Potentials under the Ricci Flow
    J. Funct. Anal., Vol. 255 (4): 1024-1038, 2008.
    abstract, pdf


  8. Backward Ricci Flow on Locally Homogeneous Three-manifolds
    (with Laurent Saloff-Coste) Comm. Anal. Geom., Vol. 12 (2): 305-325, 2009.
    abstract, pdf


  9. Differential Harnack Estimates for Time-dependent Heat Equations with Potentials
    (with Richard Hamilton) Geom. Funct. Anal., Vol. 19 (4): 989-1000, 2009.
    abstract, pdf


  10. Cross Curvature Flow on Locally Homogeneous Three-manifolds (II)
    (with Laurent Saloff-Coste) Asian J. Math., Vol. 13 (4): 421-458, 2009.
    abstract, pdf


  11. The Backward Behavior of the Ricci and Cross Curvature Flows on SL(2,R)
    (with John Guckenheimer and Laurent Saloff-Coste) Comm. Anal. Geom., Vol. 17 (4): 777-796, 2009.
    abstract, pdf


  12. Gradient Estimates for the Heat Equation under the Ricci Flow
    (with Mihai Bailesteanu and Artem Pulemotov) J. Funct. Anal., Vol. 258 (10): 3517-3542, 2010.
    abstract, pdf


  13. On Locally Conformally Flat Gradient Shrinking Ricci Solitons
    (with Biao Wang and Zhou Zhang) Communications in Contemporary Mathematics, Vol. 13 (2): 1-14, 2011.
    abstract, pdf


  14. Estimate and Monotonicity of the First Eigenvalue under Ricci Flow
    (with Songbo Hou and Jun Ling) Mathematische Annalen, Vol. 354 (2): 451-463, 2012.
    abstract, pdf


  15. Differential Harnack Estimates for Parabolic Equations
    (with Zhou Zhang) Proceedings of Complex and Differential Geometry, 87-98, 2011.
    abstract, pdf


  16. The Conjugate Heat Equation and Ancient Solutions of the Ricci Flow
    (with Qi S. Zhang) Advances in Mathematics, Vol. 228 (5): 2891-2919, 2011.
    abstract, pdf


  17. Curvature Pinching Estimate and Singularities of the Ricci Flow
    Comm. Anal. Geom., Vol. 19 (5): 975-990, 2011.
    abstract, pdf


  18. Differential Harnack Estimates for a Nonlinear Heat Equation
    (with Ben Fayyazuddin-Ljungberg and Bowei Liu) J. Funct. Anal., Vol. 265 (10): 2312-2330, 2013.
    abstract, pdf


  19. Harnack Estimates for the Endangered Species Equation
    (with Mark Cerenzia and D. Kazaras) Proc. Amer. Math. Soc., 2014.
    abstract
    , pdf


Preprints

  1. The Weyl Tensor of Gradient Ricci Solitons
    (with Hung Tran) 2013.
    abstract
    , pdf


  2. Mean Value Inequalities and COnditions to Extend Ricci Flow
    (with Hung Tran) 2013.
    abstract
    , pdf


Unpublished Manuscript

  1. Einstein Four-manifolds of Three-Nonnegative Curvature Operator
    (with Peng Wu) 2013.
    abstract
    , pdf


Lecture Notes

  1. The Hamilton Lecture: Geometrization by Ricci flow
    pdf