Local monotonicity and mean value formulas for evolving Riemannian manifolds

@inproceedings{Ecker2006LocalMA,
  title={Local monotonicity and mean value formulas for evolving Riemannian manifolds},
  author={Klaus Ecker and Dan Knopf and Lei Ni and Peter Topping},
  year={2006}
}
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. 

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