My Research

Research interests :

• Differential Geometry
• Geometric Analysis
• Nonlinear Parabolic Equations

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 and Preprints

• Isoperimetric Estimate for the Ricci Flow on S^2 x S^1. Comm. Anal. Geom., Vol. 13(4): 727-740, 2005. pdf
  
  Abstract: In this paper, we study the singularities of the Ricci flow in dimension three. If the manifold is topologically
  S^2 x S^1 , with a warped product metric, we first derive the evolution equation of the isoperimetric ratio on the base
  manifold S^2 , then we obtain a positive lower bound for the isoperimetric ratio. This shows that the dilation limit of
  the Ricci flow can not be the \Sigma x R^1, where \Sigma is the cigar soliton on R^2.

• Eigenvalues of (-\Delta +R/2) on Manifolds with Nonnegative Curvature Operator. Mathematische Annalen,
  Vol. 337 (2): 435-441, 2007. pdf
  
  Abstract: In this paper, we study the eigenvalues of operator (-\triangle +R/2) on manifolds with nonnegative
  curvature operator. We first show that all eigenvalues are non-decreasing under the Ricci flow. Then we show
  that there is no compact steady Ricci breather with nonnegative curvature operator other than the Ricci-flat one.
  

• Compact Gradient Shrinking Ricci Solitons with Positive Curvature Operator. Journal of Geometric Analysis,
  Vol. 17 (3): 451-459, 2007. pdf
  
  Abstract: In this paper, we study the following conjecture of Hamilton: Any compact gradient shrinking Ricci
  soliton with positive curvature operator must be Einstein. We first derive several identities. Then we show
  that the conjecture is true under an additional condition. Furthermore, such a soliton must be of constant curvature.

• Dimension Reduction under the Ricci Flow on Manifolds with Positive Curvature Operator. Pacific J. Math.,
  Vol. 232 (2): 263-268, 2007. pdf
  
  Abstract: In this paper, we study the dilation limit of solutions to the Ricci flow on manifolds with nonnegative
  curvature operator. We first show that such a dilation limit must be a product of a compact ancient Type I
  solution of the Ricci flow with flat factors. Then we show under the Type I normalized Ricci flow, the compact
  factor has a subsequence converge to a Ricci soliton.

• Cross Curvature Flow on Locally Homogeneous Three-manifolds (I) (joint with Yilong Ni and Laurent Saloff-Coste).
  Pacific J. Math., Vol. 236 (2): 263-281, 2008. pdf
  
  Abstract: Chow and Hamilton conjectured the long time existence and convergence of cross curvature flow on
  closed three-manifolds with negative sectional curvature. In this paper, we study the negative cross curvature
  flow on locally homogenous three-manifolds. We describe the long time behavior of these flows in each case.

• First Eigenvalues of Geometric Operators under the Ricci Flow. Proc. Amer. Math. Soc.,
  Vol. 136 (11): 4075-4078, 2008. pdf
  
  Abstract: In this paper, we prove that the first eigenvalues of $-\Delta + cR$ ($c\geq \frac14$) is nondecreasing
  under the Ricci flow. We also prove the monotonicity under the normalized flow for the case $c=1/4$, and $r\le 0$.

• Cross Curvature Flow on Locally Homogeneous Three-manifolds (II) (joint with Laurent Saloff-Coste).
  preprint, 2008. pdf
  
  Abstract: In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We
  describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic
  behavior of the negative cross curvature flow to describe the two sided behavior of maximal solutions of the
  cross curvature flow on locally homogeneous 3-manifolds. We show that, typically, the positive cross curvature
  flow on locally homogeneous 3-manifold produce an Heisenberg type sub-Riemannian geometry..

• On Locally Conformally Flat Gradient Shrinking Ricci Solitons (joint with Biao Wang). preprint, 2008. pdf
  
  Abstract: In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally
  conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral
  identity to complete noncompact gradient Ricci solitons, under the conditions of bounded nonnegative
  Ricci curvature and the Riemannian curvature tensor has at most exponential growth. As an consequence
  of this identity, we classify complete locally conformally flat gradient Ricci solitons..

• Differential Harnack Estimates for Backward Heat Equations with Potentials under the Ricci Flow.
  J. Functional Anal., Vol. 255 (4): 1024-1038, 2008. pdf
  
  Abstract: In this paper, we derive a general evolution formula for possible Harnack quantities. As a
  consequence, we prove several differential Harnack inequalities for positive solutions of backward
  heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow.
  We shall also derive Perelman's Harnack inequality for the fundamental solution of the conjugate
  heat equation under the Ricci flow.

• Differential Harnack Estimates for Time-dependent Heat Equations with Potentials (joint with Richard Hamilton).
  Geom. Funct. Anal., to appear. pdf
  
  Abstract: In this paper, we prove a differential Harnack inequality for positive solutions of
  time-dependent heat equations with potentials. We also prove a gradient estimate for the
  positive solution of the time-dependent heat equation..

Lecture Notes

• The Hamilton Lecture: Geometrization by Ricci flow. pdf