Asian Journal of Mathematics
Volume 11 (2007)
Mean Value Theorems on Manifolds
Pages: 277 – 304
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.
Green's function; mean value theorem; heat spheres/balls; Ricci flow; local regularity theorem
2010 Mathematics Subject Classification