Curvature Estimates for the Ricci Flow II

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of the norm of the Riemann curvature tensor. They also serve as characterizations of blow-up singularities.Comment: 20 page

The logarithmic Sobolev inequality along the Ricci flow in dimension 2

In this paper we present our results on the logarithmic Sobolev inequality along the Ricci flow in dimension 2.Comment: 11 page

The logarithmic Sobolev inequality along the Ricci flow

We derive a logarithmic Sobolev inequality along the Ricci flow without any restriction on time, which depends only on the initial metric via rudimentary geometric data, assuming only that a certain first eigenvalue is positive. As a consequence we obtain a uniform Sobolev inequality along the Ricci flow without any restriction on time. One application of it is a uniform kappa-noncollapsing estimate which holds true for all time. We also obtain similar results for bounded time without assuming the eigenvalue condition. The results extend to the Ricci flow with surgeries.Comment: One appendix is added. A theorem on the Sobolev inequality along the Ricci flow with surgeries of Perelman is added. Two nonlocal Sobolev inequalities are also added. These results were obtained at the time of the posting of the first version of the paper. They were originally planned as parts of two upcoming papers of the autho

The Log Entropy Functional Along the Ricci Flow

In this paper we introduce the log entropy functional and establish its monotonicity along the Ricci flow. One consequence of it is the monotonicity of the logarithmic Sobolev constant along the Ricci flow.Comment: A typo in the monotonicity inequality for the log entropy functional is corrected. Namely the factor 1/(16 \omega) should be (4 \omega)/n. (From the given computations it is obvious that this number should be (4\omega)/n.

The logarithmic Sobolev inequality along the Ricci flow: the case λ0(g0)=0\lambda_0(g_0)=0

We extend our previous results on the logarithmic Sobolev inequality along the Ricci flow in the case $\lambda_0(g_0)>0$ to the case $\lambda_0(g_0)=0$.Comment: 6 page

On the l-Function and the Reduced volume of Perelman II

In this paper we present a major application of the l-function and the reduced volume of Perelman, namely their application to the analysis of the asymptotical limits of kappa solutions of the Ricci flow.Comment: This paper has been available at the author's website. We post it here to make it easier to access. To appear in Transactions of American Mathematical Societ

Curvature Estimates for the Ricci Flow I

In this paper we present several curvature estimates for solutions of the Ricci flow which depend on smallness of certain local integrals of the norm of the Riemann curvature tensor.Comment: 22 page

Equivariant and Bott-type Seiberg-Witten Floer Homology: Part II

We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first playing the most important role. These versions are shown to be equivalent to each other. A few typos are removed.Comment: 41 page

Sobolev Inequalities, Riesz Transforms and the Ricci Flow

In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the $W^{1,2}$ Sobolev inequality along the Ricci flow to $W^{1,p}$ and $W^{2,p}$ Sobolev inequalities for general p.Comment: 23 page

Entropy Functionals, Sobolev Inequalities and kappa-Noncollapsing Estimates along the Ricci Flow

In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates along the Ricci flow, including the Ricci flow with surgeries.Comment: This survey was written for the 4th International Congress of Chinese Mathematicians to be held in Hangzhou, China in December 2007, and for the memorial volume dedicated to the 771 class of mathematics of China University of Science and Technolog