相关论文: Eigen-value monotonicity for the Ricci-Hamilton fl…

This paper has been withdrawn.

This paper has been withdrawn by the author.

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct…

The paper was removed.

This paper is incorrect, and has been withdrawn by the author.

This paper has been withdrawn.

This paper has been withdrawn due to that the same conclusion has been proposed before.

This paper has been withdrawn by the author.

This paper withdrawn since it has been published in an updated version in Biophysical Reviews and Letter Vol. 2, No. 2 (2007) 155-166

This paper has been withdrawn by the author. A much more revised version is available as arXiv:0909.4238.

This paper has been withdrawn.

This paper has been withdrawn by the author, due to a crucial error in page 5.

This paper has been withdrawn by the authors.

By decision of the first author, the paper will no longer be available.

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and…

This paper has been withdrawn by the author. Improved versions (arXiv:1109.5548 and arXiv:0708.4190) are accepted.

The paper has been withdrawn by authors. The issues studied in this paper were changed so much that we have published a new paper considering these issues. See hep-th/0406074

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.

This paper has been withdrawn.

This paper has been withdrawn by the author due to an error in section 7. There is a new version: arXiv:1011.3352.