Mean Value Theorems on Manifolds

We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as the results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation a mean value theorem with respect to 'heat spheres' is proved for heat equation with respect to evolving Riemannian metrics via a spacetime consideration. Some new monotonicity formulae are derived. As applications of the new local monotonicity formulae, some local regularity theorems concerning Ricci flow are proved.

Keywords

Green's function, mean value theorem, heat spheres/balls, Ricci flow, local regularity theorem

2010 Mathematics Subject Classification

58J35

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Published 1 January 2007