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.
     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.
     pdf

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

  20. Mean Value Inequalities and COnditions to Extend Ricci Flow
    (with Hung Tran) Math. Res. Lett., 2015.
    abstract    , pdf

  21. Harnack Estimates for Conjugate Heat Kernel on Evolving Manifolds
    (with Hongxing Guo and Hung Tran) Math. Z., 2015.
    pdf

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

  23. Differential Harnack Estimates for Fisher'ss Equation
    (with Bowei Liu, Ian Pendleton and Abigail Ward) Pacific J. Math. Soc., 2017 pdf

Preprints

  1. Einstein 4-manifolds of Pinched Sectional Curvature
    (with Hung Tran) 2016.
    abstract    , pdf

  2. Three-dimensional noncompact k-solutions that are Type I forward and backward
    (with Ben Chow and Yongjia Zhang) 2016.
    pdf

  3. A Note on the almost one half holomorphic pinching
    (with Bo Yang)., 2017.
    pdf

  4. A Differential Harnack Esitmate for the Newell-Whitehead Equation
    (with Derek Booth and D. Kazaras)., 2017.
    pdf

Unpublished Manuscript

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