  We derive several mean value formulae on manifolds, generalizing the
classical one for harmonic functions on Euclidean spaces as well as later
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 space-time
consideration. Some new monotonicity formulae are derived. As applications of
the new local monotonicity formulae, some local regularity theorems concerning
Ricci flow are proved.
