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Eigenvalues and energy functionals with monotonicity formulae under Ricci flow

Junfang Li
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Abstract

In this note, we construct families of functionals of the type of $\mathcal{F}$-functional and $\mathcal{W}$-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using these new functionals, we also give a new proof of Perelman's no non-trivial expanding breather theorem. Furthermore, we prove that compact expanding Ricci breathers must be Einstein by a direct method. In this note, we also extend X. Cao's methods of eigenvalues\cite{C} and improve their results.

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Junfang Li. 2007-01-26. Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. https://arxiv.org/abs/math/0701548

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