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We derive identities for general flows of Riemannian metrics that may beregarded as local mean-value, monotonicity, or Lyapunov formulae. Thesegeneralize previous work of the first author for mean curvature flow and othernonlinear diffusions. Our results apply in particular to Ricci flow, where theyyield a local monotone quantity directly analogous to Perelman's reduced volumeV and a local identity related to Perelman's average energy F.

Local monotonicity and mean value formulas for evolving Riemannian manifolds