Eigenvalues and energy functionals with monotonicity formulae under Ricci flow

By Junfang Li. 2007

95 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0bd0caada400a1615f5433fd92fc11b3bafce84c

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

ENTROPY AND LOWEST EIGENVALUE ON EVOLVING MANIFOLDS (2013, methodology): “We thus present a unified formula of various W entropies established by various authors for different flows (including the static case), see [5, 8, 9, 11, 12, 13].”
FIRST EIGENVALUES OF GEOMETRIC OPERATORS UNDER THE RICCI FLOW (2007, methodology): “[Li07b] Junfang Li.”
Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation☆ (2009, result): “Here, we will present a direct proof which is based on a change of variable argument, see similar argument in [14].”