Asian Journal of Mathematics

Mean Value Theorems on Manifolds

Lei Ni

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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.

Article information

Asian J. Math., Volume 11, Number 2 (2007), 277-304.

First available in Project Euclid: 31 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J35: Heat and other parabolic equation methods

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


Ni, Lei. Mean Value Theorems on Manifolds. Asian J. Math. 11 (2007), no. 2, 277--304.

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