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Monotonicity Formulas of Eigenvalues and Energy Functionals Along the Rescaled List’s Extended Ricci Flow

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Abstract

In this paper, we study monotonicity formulas of eigenvalues of Laplacian and entropies along the rescaled List’s extended Ricci flow, which is a weakly coupled system of second order and the motivation to study this flow stems from its connection to general relativity. The rescaled List’s extended Ricci flow, which is a generalized version including the extended Hamilton normalized flow, is equivalent to the List’s extended Ricci flow up to a homothetic rescale of the metric at each time. Moreover, we find natural entropies of \(\mathcal {F}_k\)-functional and \(\mathcal {W}_k\)-functional and show that they are monotonic.

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Acknowledgements

The authors would like to express their deeply gratitude to the referee for valuable comments, which make the paper more readable.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, People’s Republic of China

    Guangyue Huang

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Zhi Li

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  1. Guangyue Huang
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  2. Zhi Li
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Correspondence to Guangyue Huang.

Additional information

The research of the first author was supported by NSFC Nos. 11371018, 11671121.

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Huang, G., Li, Z. Monotonicity Formulas of Eigenvalues and Energy Functionals Along the Rescaled List’s Extended Ricci Flow. Mediterr. J. Math. 15, 63 (2018). https://doi.org/10.1007/s00009-018-1105-0

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  • DOI: https://doi.org/10.1007/s00009-018-1105-0

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