Heatball localizations of monotone functionals on evolving manifolds

Monotone functionals give important information for the analysis of geometric evolution equations. I will describe a quite general mechanism for localizing such functionals. This construction results in what may be regarded as local mean-value, monotonicity, or Lyapunov formulas. In particular, the construction yields a purely local monotone quantity directly analogous to Perelman's reduced volume for Ricci flow, and another related to Perelman's average energy. This is joint work with Klaus Ecker, Lei Ni, and Peter Topping.