  We derive identities for general flows of Riemannian metrics that may be
regarded as local mean-value, monotonicity, or Lyapunov formulae. These
generalize previous work of the first author for mean curvature flow and other
nonlinear diffusions. Our results apply in particular to Ricci flow, where they
yield a local monotone quantity directly analogous to Perelman's reduced volume
V and a local identity related to Perelman's average energy F.
