Eigenvalues and energy functionals with monotonicity formulae under Ricci flow

• Autores: Jun-Fang Li
• Localización: Mathematische Annalen, ISSN 0025-5831, Vol. 338, Nº. 4, 2007, págs. 927-946
• Idioma: inglés
• DOI: 10.1007/s00208-007-0098-y
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• Resumen

  • In this note, we construct families of functionals of the type of [F] -functional and [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.