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 Cao’s methods of eigenvalues (in Math Ann 337(2), 2007) and improve their results.
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Li, JF. Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann. 338, 927–946 (2007). https://doi.org/10.1007/s00208-007-0098-y
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DOI: https://doi.org/10.1007/s00208-007-0098-y