Related papers: Eigen-value monotonicity for the Ricci-Hamilton fl…

This is a revised version of our short note [arxiv.math.DG/0403065] where we discuss the monotonicity of the eigen-values of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. We…

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This paper is withdrawn from submission due to a critical error in the proof of main theorem.

There have been comments on this paper which point out unclear motivation and definitions on noncommutative momentum introduced. Therefore, this paper is withdrawn by the author for more clear presentation.

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In this paper, the author discusses the eigenvalues and entropies under the harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map flow. We give an alternative proof of results for compact steady and expanding…

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This paper has been withdrawn by the author(s) in the light of several other works available and due to a misunderstanding in the authorships.

In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem \ref{theo rescaled}.